In some cases, it is an example of "Just because the cat had kittens in the oven, it doesn't make them biscuits", i.e., some Scots emigrated first to Ulster, then on to America. My Witherspoon ancestors stopped in Belfast, but not for a whole generation, before heading to Carolina.

I believe the distinction came largely as described above, when variously derided Irish-Irish came to the US in great waves. Another distinction seems to be that Scotch-Irish / Scots-Irish are overwhelmingly Presbyterian, whereas the Irish-Irish have historically been Roman Catholic.

I realize Matt's article explains this more thoroughly and more clearly, but not everyone will follow the link. For those wishing to read even more, I would recommend DAVID HACKETT FISCHER'S ALBION'S SEED. Fischer calls the Ulster Scots "borderers" and describes them as one of four large groups of British immigrants, each preserving characteristic folkways.