There is not necessarily any terminology that would be used, other than the city needs to match the state in order to know WHERE the city is and for correct filing. There are multiple states with a city named New York, so the locality should match the state.
If they refuse to correct it, you can just correct it when you enter it on your return so it is reported properly on your return and so you are able to file the NYC return. You will not be able to file the NYC if you do not choose the correct state.
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