Merriam Webster's latest mode to the contrary, words and terms do have fixed meanings.

A foreigner living in America, whether he is an illegal alien, a legal resident alien, or a naturalized citizen in "the immigrant".

A child born in the US to an immigrant is a "first generation American".

In genealogy, those of us who have colonial ancestors refer to the person who moved to the colonies as "the immigrant" and count our generations from the first generation born in the colonies. For example: My immigrant ancestor moved to Jamestown in 1607. I am a 12th generation American, even though the US didn't come into being until much later.