It might be that Jerusalem is sometimes used broadly of an entire district, but when used specifically of a single City it's just Jebus. Some references to Jerusalem and Zion in the same verse often taken to verify their being synonymous, can also work as listing separate cities side by side. Like Isaiah 64:10 which says cities, plural, then lists Jerusalem and Zion. But since Zion also arguably has both a poetic broader application and a more specific one, perhaps it fits when paired with Jerusalem, two names that refer to different specific cities but basically the same area when applied broadly. Psalm 76:2 also makes sense as referring to Salem and Zion as separate cities.
It's possible sometimes Jerusalem and Zion are paired together to represent the two tribes of the Southern Kingdom, Benjamin and Judah, Psalm 78:68 says Zion is a mount of Judah. Which can in turn be taken back more broadly to represent both wives of Jacob, Benjamin from Rachel and Judah from Leah.
And perhaps David's design for this area was similar in intent to the original plan for Washington DC, taking parts of both Maryland and Virginia to create a capital District. Isaiah 24:23 refers to Yahuah ruling in Zion and in Jerusalem, as if they are separate.
One question that might pop into your mind from the idea of separating Jerusalem from the City of David is, which city then is Ariel in Isaiah 29? "Where David dwelt" could apply to both but arguably fits the City of David better. And Zion is mentioned explicitly. Also Ariel means "Lion of God", that fits it being a Judean rather then Benjamite city, as Judah is the Lion in Genesis 49. Other tribes (Gad and Dan) are associated with Lions elsewhere (Deuteronomy 33), but not Benjamin.
I think the house David built with materials provided by Hiram of Tyre was in Jebus/Jerusalem, where he lived and had children with his wives from Jerusalem, and that could be the same archeological site it's usually associated with. But the Fort of Zion was in the City of David, that fort already existed.
Ophel is a place-name linked to the Gihon once. It's a Hebrew word for Tent sometimes used of the The Tabernacle, and in the KJV is translated " tabernacle", so it may not always refer to the same place. Maybe the Ophel in the City of David could have been where David's Tabernacle was?
Only 2 Chronicles 1:4 says David pitched a Tent for the Ark in Jerusalem rather then Zion or the CIty of David. First off the books of Chronicles probably entered their final form later, so a broader definition of what qualifies as Jerusalem may make more sense there. But also this reference comes after David had purchased Moriah, so maybe the Ark did some moving around during this period.
I think Jerusalem is usually the City called the Daughter of Zion or Daughter of Sion. But "daughter of ____" can sometimes refer to a separate City that is related in some way. Like Tyre being the Daughter of Sidon/Zidon in Isaiah 23:12. And the Daughter of Babylon I think is likely Hammurabi/Nebuchadnezzar's Babylon, the daughter of the original Babel which was Eridu.
The City we usually call Jerusalem I feel is obviously the Jerusalem of the The Gospels & Acts and thus the Jebus of the Hebrew Bible. Where Solomon and Zerubabel/Herod's Temples were built. So where then is the City of David and Mount Zion?
Luke Chapter 2 in verses 4 and 11 calls Bethlehem the City of David, and endless Christian commentaries try to explain why this doesn't contradict the Hebrew Bible's City of David being Jerusalem by saying both could be described that way. Yet we're supposed to use Scripture to interpret Scripture, and Luke said "The" not "a". And Christians view the New Testament as revealing and clarifying the "Old Testament". This argument would not convince Jews or other non Christians of course, so fortunately I have some directly from the Hebrew Scriptures.
1 Samuel 20:6 when speaking of David refers to Bethlehem as "his city", that predates the exact phrase "City of David' ever occurring. When you think about it this should always have been obvious, the hometown of David is the City of David. To go back to a previous point, Bethlehem was in Judah.
If you object, "David had to capture his own home town?" remember what I said above, in my theory Zion didn't need to be captured, it was the base of operations for capturing Jebus. Bethlehem is also in a mountainous region, in fact it's elevated higher then Jerusalem. Perhaps the Gihon was the spring now known as the Spring of Etam, or Atan?
It's interesting to note that the Crusaders also captured Bethlehem first, Godfrey sent Tancred to take it, then they used it as a base in their siege of Jerusalem. Lots of people overlook this detail of the Crusades, but once you're aware of how Bethlehem is elevated higher then Jerusalem, you realize it is ideal to secure that area first if you want to siege Jerusalem.
Micah 5:2 is the key Prophecy that The Messiah would be born in Bethlehem. But remember the Chapter divisions were not in the original text, and Micah 5 does sound like it's starting in the middle of something. Micah mentions Zion constantly, particularly in chapter 4.
The first time Bethlehem is mentioned it is home to the Tower of Edar in Genesis 35:20-21. Micah 4:8 refers to the Tower of Edar (Tower of The Flock in the KJV) as the Stronghold of Zion. Then later refers to Jerusalem arguably as a separate city. Some traditions say it was from the Migdol Eder that the Angel announced the Birth of Jesus to the Shepherds.
Psalm 132 mentions Ephratah in a context that seems to place the Tabernacle (Ophel) and The Ark there. And it is a Davidic Psalm. The City of David housed the Tabernacle of David and The Ark during most of David's reign. And that Psalm also uses the name Zion.
2 Samuel 2:32 says David's nephew Asahel was buried in Bethlehem in the sepulcher of his father. Kings of the House of David are repeatedly refereed to as being buried in the City of David, and resting with their fathers. Starting with David himself in 1 Kings 2:10 being buried with his father in the City of David (Acts 13:36 also says David was buried with his fathers). And it turns out Bethlehem does have a site with a tradition of being where David was buried. Or the Kings might have been among those buried in the Bronze Age caves built where modern Efrat is.
https://en.wikipedia.org/wiki/David's_Tomb " In the 4th century CE, he and his father Jesse were believed to be buried in Bethlehem. The idea he was entombed on what was later called Mt Zion dates to the 9th century CE." Rabbi Dr. Ari Zivotofsky, 'Where is King David Really Buried?,' The Jewish Press, May 15th 2014. "By the mid-fourth century, the tombs of King David and his father, Jesse, are described as being in Beit Lechem.[See Limor, “King David’s Tomb.”] The first mention of Mount Zion as King David’s final resting place was in the ninth century". Back to Wikipedia "4th century Pilgrim of Bordeaux reports that he discovered David to be buried in Bethlehem, in a vault that also contained the tombs of Ezekiel, Jesse, Solomon, Job, and Asaph, with those names carved into the tomb walls."...[Ora Limor, "The Origins of a Tradition: King David's Tomb on Mount Zion," Traditio 44 (1988): 459.] "Having initially revered David's tomb in Bethlehem, Muslims began to venerate it on Mount Zion instead but no earlier than the 10th century following the Christian (and possibly Jewish) lead. In the twelfth century, Jewish pilgrim Benjamin of Tudela recounted a somewhat fanciful tale of workmen accidentally discovering the tomb of David on Mount Zion."Asahel was a maternal Nephew which makes the above argument not quite a slam dunk exactly. But his father is never identified. And all three of Zeruiah's children are called sons of Zeruiah rather then by their father. That makes it possible they may have been born out of wedlock and so mostly treated as part of Jesse's family. Either way being buried in Bethlehem means, if it was his direct father he was buried with, he was one from the same city and so probably at least the same Tribe. That David's nephews were so important to him means he may have insisted they be buried as part of the royal family.
Some kings are assumed to not be buried with the others in the City of David however. Manasseh and Amon were buried in the Garden of Uzza or Uzzah, in 2 Kings 21. Manasseh is still said to have "slept with his fathers", however that terminology is arguably more vague being sometimes just used of death in general. But, Uzzah was also the name of the person who died from touching the Ark as it was transported to the City of David, and David named a location after this Uzzah, Perezuzzah. And another Uzza is listed in 1 Chronicles 6:29 as a Levite who was appointed a Musician in the Tabernacle of David. So the name of Uzza can be linked to the City of David.
Jehoram was buried in the City of David but not with the other kings because of the condition he died in according to 2 Chronicles 21:20. 2 Chronicles 24:25 has a similar situation with Joash. Jehoiada, a priest who married Jehosheba, a daughter of Jehoram, is refereed to as being buried among the Kings in the City of David in 2 Chronicles 24:16. So that adds more context to the Asahel situation.
Another King explicitly said not to be Buried with the others was Ahaz in 2 Chronicles 28:27, and this time it doesn't mention the City of David but says he was buried in Jerusalem. Maybe where he was buried could be a clue to Manesseh and Amon's Garden of Uzza.
The name of Uzza/Uzzah here could be a variation of Uzziah, another name of King Azariah. This king was originally buried "in the field of the burial which belonged to the kings" (2 Kings 15:7; 2 Chr. 26:23), but... that leads us to the Uzziah Tablet.
In 1931 an archeological find, now known as the Uzziah Tablet, was discovered by Professor E.L. Sukenik of the Hebrew University of Jerusalem. He came across the artifact in a Russian convent collection from the Mount of Olives. The origin of the tablet previous to this remains unknown and was not documented by the convent. The inscription on the tablet is written in an Aramaic dialect very similar to Biblical Aramaic. According to its script, it is dated to around AD 30-70, around 700 years after the supposed death of Uzziah of 2 Kings and 2 Chronicles. Nevertheless, the inscription is translated, "Hither were brought the bones of Uzziah, king of Judah. Not to be opened." It is open to debate whether this tablet really was part of the tomb of King Uzziah or simply a later creation. It may be that there was a later reburial of Uzziah here during the Second Temple Period.Being buried on the Mount of Olives, is pretty interesting.
It may also be interesting to note that a Bethlehem site is believed to be where Herod was buried, the Herodion. And Herod's Official Biographer claimed him to be of Davidic ancestry. I also wonder if Herod built his Herodium fortress over the older fort David had used that I'd mentioned earlier, possibly eliminating all evidence of the older fort. Some aspects of Josephus description seem consistent with the idea of an older fortification existing there, it was the site of a battle before Herod built anything.
Bethlehem is never mentioned (by that name at least) in the books of Kings and only once in 2 Chronicles during the reign of Rehoboam. As if during the Kingdom period calling it by that name was phased out in favor of the City of David. Of course between Solomon and Hezekiah the only references to the City of David are as where the Kings were buried.
If you still think The Man-Child of Revelation 12 is Jesus. It is Zion far more often then Jerusalem refereed to as travailing in Childbirth. With my view of The Man-Child as The Church at The Rapture, maybe Bethlehem will play a role in that? After all a heavenly Mount Sion is important to Revelation 14.
Yahuah-Shammah is nine times the size of modern Jerusalem, according to the most common estimate of it's size, it could be larger. Bethlehem is about five miles south of Jerusalem. Yahuah-Shammah could be large enough to encompass both Jerusalem and Bethlehem.
But remember not all references to Bethlehem are to the one in Ephratah of Judah, there is another lesser known one in the North. Sometimes people will try to argue that is where Jesus was born, but Micah, Matthew and Luke all make qualifiers ruling out the Northern One. And the reference in John 7 would make no sense in that context if they meant a city in Galilee.
So I have run into a potential problem with my Bethlehem theory. Nehemiah 3:15-16 and 12:37 refer to the City of David as seemingly pretty clearly within the Jerusalem he rebuilt, and refers to the burial site of the Kings being there. But I shall look more into that as there could be an explanation.
The potential answer to that issue is that Nehemiah's wall was larger then we usually think it was, that it encompassed Jerusalem and Bethlehem. Maybe the wall we usually identify it with isn't it at all, or maybe it is but was only part of it. And that most of what Nehemiah built we won't find the remains of due to the conquests of Israel by Antiochus Epiphanes and Rome.
Given my speculation that the Construction projects of Suleiman The Magnificent could be a second fulfillment of the Daniel 9 Prophecy fulfilled by Nehemiah. It's interesting to note that he also built fortifications in Bethlehem, like The Castle of The Pools.
Update: a Post in which I consider there may have been two Arks, possibly filling some gaps in this study.
Further Update: The Death of Rachel and Birth of Benjamin
Is traditional assumed to have been in Bethlehem. But many have read Genesis 35 more carefully as saying the birth of Benjamin, Death of Rachel and her Burial were on the way to Ephratah and the Migdal Eder from Bethel.
If so that makes it likely these events happened in land later allotted to Benjamin. As the only of the 12 sons born in the Land, perhaps it makes he'd be allotted his birth place. 2 Samuel 10:2 refers to her Tomb as being in Benjamin. And there are traditions saying it is specifically in Ramah, which could be relevant to the "Voice Crieth form Ramah" Prophecy of Jeremiah 31:15 quoted in Matthew 2.
As far as my citing it as evidence of Bethlehem being Zion. I note that this argument observed that after these events Jacob traveled past the Migdal Eder and set up a Tent. Perhaps this Tent was where the Tabernacle of David was later sent up? Maybe that is the origin of the site popular viewed as Rachel's Tomb today?