Was there a 21st precinct in New York City?
There was no 21st Precinct Station in NYC. There was (and is) a 20th and 22nd, but no 21st. It was precisely because there was no 21st Precinct in real life that Sidney Kingsley set his stage play DETECTIVE STORY in the detective squadroom of the 21st Precinct, and that the radio drama, 21ST PRECINCT, WA set there.
If you like West Side Story, the police were from the 21st Precienct.
New York City is the largest city in New York State. New York State is #11 of the 50 States of the United States of America. Note that while New York City is the largest city in New York State (as well as the largest city in the country), it is NOT the state capital. The capital of New York State is Albany.
The official name is just "New York." Their web site uses "New York City" or the initials, NYC. But you can use either, both "New York City" or "City of New York" are valid when referring to the city which name is New York located in the state of New York. See their link below. Commonly known as the Big Apple, the NYC, New Yawk.