The unification thing has ebbed back and forth over the years.

There was a time when being British was stressed, when the word "Scotland" itself was often dropped and was replaced by the term "North Britain".

In 1708 the Royal Scotch Dragoons' (sic) title was changed to the Royal North British Dragoons. "North British" was not replaced by "Scots" until 1877.

Likewise the Scotch Fuzileers' (sic) title was changed to the North British Fuzileers in 1707, the "Scots" not returning until 1877.

But now Scottishness is stressed once more and Scotland has a brand-spankin' new (and unspeakably hideous) Parliament building.

About the term "Scotch-Irish" it was originally used to refer to the Ulster Scots. Many of the Scotch-Irish who immigrated here were only a generation removed from Scotland, and were not Irish in the least, though they arrived here from Irish ports. (Hence the confusion amongst Americans of that time: these people arrived from Ireland, they must be Irish.)

There was no mixed Scottish/Irish culture here at any early date. The mass of Ulster Scots arrived here more than a hundred years before the Potato Famine Irish began arriving, and by that time the Ulster Scots had mostly become Methodists and Baptists and were largely melded into British Protestant America. (Yes in Appalachia the descendants of the Ulster Scots continued to make whisky and fiddle and dance in the old Scottish way.)

The early 18th century Ulster Scots pushed to what was the frontier at that time; the mid 19th century Potato Famine Irish went to the big Northern cities. The former produced politicians such as George Wallace, the latter politicians like JFK, two men quite different in their speech, religion, and culture in general.