Random thoughts:

Ok, I don't think the question is whether the term is ambiguous anymore...that seems to be established. But this state of affairs does not sit well. Some resolution would be useful. If you are just talking to someone it is easy enough to ask them to clarify but what if it is just written or an interview on TV? Or what if it is a class situation as one person was asking about. What if the teacher asks, gets an answer, and then just assumes they are on the same page and does not ask for a clarification, then anyone listening would not know which it was either.

Here's how we could do it: instead of the whole "born here" thing I think it is more important were you were raised. If you were the first generation raised here then call that first generation if you have parents or grandparent who came here just call them generation zero or immigrants. It is kinda strange how immigrant can sound negative in a land of immigrants. If for instance someone asked “Are you or your parents immigrants?” It is hard to not take it negatively.

You can say in response to "Are you first generation or second?" with “I am the proud child/son/daughter of immigrants,” or “I immigrated in 1988” or whatever year, or “my parents came-here/immigrated in 1972” or whatever or if you want to be less specific you can say “…in the 1970’s.” You don't actually have to say 1st or 2nd; the important thing is to communicate clearly. They still may ask “Were you born here?,” if you tried the first suggestion. You can choose to answer that or not. If you grew up here, I would just say that. The other stuff, that’s none of their business unless they are officials or something. It’s like asking your birthday.

The whole generation thing is not applicable if your parents are different generations in my opinion. Or if it is, it would be based on the earliest ancestor in America not the most recent or the average or the male line or what have you unless the earliest is native. If they are and they are the only one and generations ago it would be silly to call yourself Native American. And it would be quite a chuckle elicitor if you looked Korean and spoke Korean and said you were Native American. And it would be a real head scratchier if you said I am Korean but my family has been here for thousands of years.

In genealogy circles 1st generation is usually the first generation to have or raise children here, or whatever place we are talking about. So if they drug their parents or grandparents here the son or daughter who actually had a family here would be 1st generation.

First generator? First generated? What about that? First progenitor? Too bad you can’t use “colonist” or “pioneer” they sound nice but it only applies to the first settlers from abroad. The options go downhill from there.

Well, that doesn’t seem to clarify much of anything. I stupidly thought I might get somewhere.

Hey what about “Yes” as the answer to “Are you first or second generation?”?
Don’t know why punctuation seems to be such a nightmare. Sorry, if I damaged all the grammarians’ retinas.