American here. The way we use the word "immigrant" is that if you were born here and grew up here, you are not considered an immigrant.

Unless you're not white, in which case you're called an "anchor baby" or "second-generation immigrant".

This is how Europeans (and maybe not only us) commonly think about Americans, all immigrants from Europe first, from the rest of the world next. However probably it's not how they think about themselves. I'd like to hear from them.