The Netherlands is the official name of our country.
Foreigners often use the name Holland, which refers to the largest and richest province our country had during the Golden Age (17th century). People from other countries trading with e.g. the VOC (East-Indian trading Company) referred to the Netherlands as Holland, because every merchant they encountered came from that region. The name stuck throughout history, but it's not official.
New York used to be New Amsterdam.
and now it's called new york
It was once called New Amsterdam.
New York
New York was once called "New Amsterdam" because the Dutch owned New York.
Before we bought New York (once called New Amsterdam), it was a Dutch colony.
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.)
They called it: Nieuw Amsterdam (New Amsterdam)
The original name of New York is New Netherlands.
They called it: Nieuw Amsterdam (New Amsterdam)
New York City (not the state) used to be called New Amsterdam. It was named many names (Dutch and British) over the years prior to our winning the American Revolutionary War.
New Amsterdam