Good discussion. I'm not scholar either, but from my understanding Jerusalem also was called the city of David:
2 Samuel 5
6 And the king and his men went to Jerusalem against the Jebusites, the inhabitants of the land, who spoke to David, saying, “You shall not come in here; but the blind and the lame will repel you,” thinking, “David cannot come in here.”
7 Nevertheless David took the stronghold of Zion (that is, the City of David)...
9 Then David dwelt in the stronghold, and called it the City of David. And David built all around from the Milloand inward.
10 So David went on and became great, and the Lord God of hosts was with him.
As king, David finally was able to bring the Ark of the Covenant into Jerusalem:
2 Samuel 6:
12 Now it was told King David, saying, “The Lord has blessed the house of Obed-Edom and all that belongs to him, because of the ark of God.” So David went and brought up the ark of God from the house of Obed-Edom to the City of David with gladness.
Hebrew scholars believe Bethlehem to be the City of David, and where he was anointed king. God sent Samuel there to annoint him as the chosen one (1 Samuel 16):
And Samuel said to Jesse, “Send and bring him. For we will not sit down till he comes here.” 12 So he sent and brought him in. Now he was ruddy, with bright eyes, and good-looking. And the Lord said, “Arise, anoint him; for this is the one!” 13 Then Samuel took the horn of oil and anointed him in the midst of his brothers; and the Spirit of the Lord came upon David from that day forward. So Samuel arose and went to Ramah.
According to 2 Samuel 2: 1-4, David was anointed in Hebron as king of Judah and then again as King of all Israel by the elders in 2 Samuel 5:1-5 ; but with God doing the choosing, make sense he had already been anointed king of all Israel in Bethlehem as a kid.