Obviously a confusing term but a long and at times amusing thread (particularly the time travel comment)...

As a "7th generation" Australian child growing up in Australia my dad would refer to some of my friends who were the children of Italian immigrants as "second generation"... given it was confusing way back then as a kid, I have been paying attention to the usage of this phrase for about 35 years :). And based on all these years of listening to the usage of this term I'd say 80% of the people who have used it used it this way (the same as the US census... immigrants are first generation, first born are second generation... so if we were to cast votes, maybe that would satisfy everyone? doubt it).

If the whole family, grandparents, parents and young kids move from their mother country... all bets are off :) But I would tend to think the foreign-born young kids would be 1st generation, and the first-born children would be "second generation" citizens of their new country. Thus cementing "second generation" as the first generation born in the new country. Just to make it completely lucid.

As for why some people don't want to be naturalized, there are many, many good reasons not to become a US citizen... such as the IRS. But ultimately I think a lot of people like to move around without giving up their cultural identity... and basically want to enjoy a life without borders, which really are artificial impediments to human exploration and freedom.

In the end it doesn't really matter, you are who you are and we are all children in the eyes of the universe.