when i visited New York i saw that they used the word 'precinct' and I thought the word precinct meant

Question

'district'

So does it refer to the district AND the police station?

Answer 1

Yes -- that is also the definition in the dictionary.

Answer 2

It does. In cities the size of NY, they have many police station buildings. Each is in the "district" they cover. They call their building a precinct. Most have numbers denoting them.

Answer 3

You know, just like their recently indicted (for pleading guilty to accepting bribes) former commissioner of the NYPD, their way of doing things is the only way ;-)