I think it is a 4-level response:
1. If I am within Queens, I say "Woodhaven". (When I was a kid, the answer was "QV" -- usually not "Queens Village", but "cue vee".)
2. If I am in another borough, or in nearby areas like North Jersey, Long Island, or Westchester, then the answer is "Queens".
3. If I am talking to someone from elsewhere within the Northeast, I say "New York City". (That's "New York City" as opposed to "New York" so that they don't mistake "New York" for the state, and ask "what city?")
4. When talking to anyone else, I say "New York", and that is always understood to mean the City.
Ferdinand Cesarano