Here's the correct timeline:
* 1624: The Dutch West India Company establishes a trading post on Manhattan Island, which they called New Amsterdam. This was the beginning of Dutch colonization in the area.
* 1626: The Dutch establish settlements in the area now known as New Jersey, part of the broader New Netherland colony.
* 1664: The English capture New Amsterdam and rename it New York after the Duke of York.
So, while New Amsterdam was the first Dutch settlement in the area, it became part of New York, not the other way around.