first-gen•er•a•tionIf definition 1 is correct, then definition 2 is incorrect, and immigrants are zeroth generation.
1. being the first generation of a family to be born in a particular country.
2. being a naturalized citizen of a particular country; immigrant
I think that the problem here is that most people do not understand the zero. If the children of immigrants are first generation, then they don't know what to call the parents because they don't understand that a number can be less than one.
Google reports 1000s of web pages using the term first generation immigrant, even tho there is no such thing. It has almost as many for second generation immigrant, and that is also a nonsense term.
Only one definition makes sense. The folks who permanently move to the USA are immigrants, and generation zero. The first generation born in the USA are called first generation; they are not immigrants. The children of the first generation are called second generation, and so forth.