[johaugum]

“The city” refers to Manhattan specifically, not New York City in general. Being in the other boroughs, you might say “I’m heading to the city later”.

[Trasmatta]

Or if you're upstate and going to one of the boroughs, you'll generally refer to it by name. "I'm going to Brooklyn / Queens / the Bronx", not "I'm going to the city" or "I'm going to New York City".

[detourdog]

I see what you are saying but one could easily go to two or more of those boroughs in single trip.