Ah yes, Lace Curtain Irish—I’m familiar with the term, but the only person I’ve ever heard use it is my brother, and we’re second-generation Americans.
All four of my grandparents were born and raised in Ireland, but I’ve never thought of myself as anything but American.
The first two categories integrate into American life quite well over time (I have some of both in my background), but the last category stay brutish and nasty, no matter how much money they get and how much education they're exposed to. I think the Kennedys qualify.