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

So, they may live in Cupertino, Palo Alto, or Oakland, but when they say “we’re going to The City for a concert on Saturday!” we can hear them pronounce those capital letters as an unambiguous reference to the one, true City.

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

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.