He like so many others including Reagan are Irish-Americans, not Irish.
I think you're onto something with the Irish-American specification.
Tip O'Neill was Irish-American in the sense of having been raised in the US among Irish immigrants and their descendants, and imbuing much of that culture, which would have included, among other things, a close familarity with (if not always adherence to) the Catholic faith.
Can the same really be said about Ronald Reagan?