Formally, it's known as New York, New York. The main part most people think of as NYC is formally known as Manhattan.
Of course, formerly it was known New Amsterdam. (The tense of your question suggests you meant formerly and not formally. Learn to spell.)
It was once called New Amsterdam.
New York used to be New Amsterdam.
New York was once called "New Amsterdam" because the Dutch owned New York.
It is now called New York.
New York City
No the city was once New Amsterdam in New Netherlands.
New York City is a part of New York State.
New York City.
New York New York New York
New York City
New York City
New York City is located at the southern edge of New York State. It is sometimes called, "the City of Southern New York."