LA is just called LA, though people do say downtown LA. I've only heard of San Francisco being referred to as the City by West Coasters.

Similarly around Salt Lake City - you may go "downtown" but not to "the City".

It probably has to do with the density gradient. With more spread out urban areas having less distinction between "the city" and outlying areas.

That's because people don't think of LA as a city, just a slightly more urbanized area within the larger semi-suburban expanse of the metro area.

It's all city down there...that's why.