This term is not ambiguous just misunderstood it seems, there are actually 2 ways to be a "First Generation American", allow me to explain:

1. The more popular method of becoming one would be to be an immigrant who comes here legally and becomes a naturalized citizen, thus being the "First Generation" of their family line to be an "American".

2. The other way, which has become in recent decades more popular and accepted, for obvious political reasons, if people come here illegally and never seek citizenship or get deported, if they have a child, that child becomes the "First Generation" of their family to become an "American" because being born in America naturally makes you an American citizen, whereas the parents of that child are illegal aliens and they might be a generation before their child, they are not Americans in any legal sense.