this is the GIS site of the muni, you will find data for the entire city
https://jergisng.jerusalem.muni.il/baseWab/?config=../gisviewerngsupport/api/InjectingConfig&locale=en

Here is the official site of Jerusalem Municipality. In this page, which lists the neighborhoods of the city, you will neighborhoods from all over the city.
https://www.jerusalem.muni.il/en/neighborhoods/

Hello,
Israel annexed east Jerusalem in 1967 after the 6 days war.
Jerusalem has only one municipality, service the entire city. Theres no "east" or "west" municipalities.
This is the situation on ground, what official source you looking for?

Hello mr. bot
I have removed 6 ways ive uploded earlier, then replaced them in more updated, accurate versions.