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Today it's a half archeological site half museum.

...and all tourist trap.
Isn’t Jerusalem itself a city? Or is it more similar to New York City, with the area sectioned into boroughs? >

Jerusalem is the city itself. "City" of David isnt an actual city but an archeological site where the city was so it got "city" as a nickname.
Inside the modern city there is an area called "city of David" which was where ancient Jerusalem stood. Today it's a half archeological site half museum.
The oldest structure is dated to around 1200-1300 BC