Which one is correct?

I would say "on the west side of the city" or "in the west of the city" 3 is the only one that's absolutely incorrect, but I probably wouldn't use any of the others.

usually people would say “I’m from the west side” or “I live on the west side”

Probably the honest answer (from a UK/Aus perspective but likely in the US too) is each city has its own terminology. In Melbourne we lived in the inner-west or just Yarraville. In London you would refer to North London or West London, or the specific area you live in, "e.g. Harrow, Ealing, etc" If you were using that phrasing, though, "I live in the west side of the city", assuming someone had no understanding of that city's geography I suppose.

“I live on the west side” is how I normally hear this explained.

3 is completely wrong: you would never say "half part". All the others are valid and would be clear. 2 is probably the most natural.

2 sounds best. 1 sounds a bit unusual but understandable. 3 and 4 sound totally wrong. We do not say "the half part of" something. And "western half" sounds far better than "west half". I can't explain why "west side" and "west part" sound okay to me, but "west half" and "half part" sound bad.

On the west side

I’d say the west side. Or the western side. West side because my city is Manhattan and everyone says West Side. Western side for cities that don’t have a fixed phrase for that.

1,2, and 4 are perfectly fine. 3 is grammatically incorrect.

“I live in the western part of the city” sounds fine but probably wouldn’t be my first choice. I’m sure some people might say it that way though. 1 sounds a bit weird and 3 is really weird. “I live in the west half of the city” is also a bit weird. But I think a lot of people would also say “I live on the west side of the city”

1 and 2 are acceptable. You could also say "I live on the west side of the city."

all of them work except for 3