I live in a city (a quite small one) and if I say that I'm going into the city, then it means I leave my neighbourhood (it's in the same city) and I'm going to the city centre.

Same. I'm in Berlin, I may say "I'm going to the city/town/village" for different parts of the city-state.

Same with NYC, "going to The City" usually just means going to Manhattan.

The city of London literally has a city also named London inside it and yeah, "Where are you going?" "The City" means clearly that tiny part. There is another small older city inside London, Westminster, but if you meant Westminster you'd say "Westminster" never "The City" in general.

I grew up equidistant from Philadelphia and New York. "The city" was exclusively used for New York regardless of the many cities between us

> In the San Francisco Bay Area, the locals refer to SF proper as “The City” (not “Frisco”.)

Eh, they also say “I’m going to SF”

But agreed definitely not “Frisco”