Scotch-Irish (or Scots-Irish) Americans are American descendants of Presbyterian and other Protestant dissenters from the Irish province of Ulster who migrated to North America during the 18th and 19th centuries.
My Great Grandmother would never use this term even though she emigrated from Northern Ireland as far as she was concerned she was a Scot who happened to have had to live in Northern Ireland before she moved to Canada and she was still a Scot when they emmigrated again in 1901 to Los Angeles and never ever contradict her.
Adventures and Musings of an Arch Druidess 
You are such a wealth of information! Here is what I know:
According to family lore, we are in some way related to John McIntosh of McIntosh apple fame. I don’t know how, or if this is even true.
My mother used to be adamant that we were IRISH! However, as you say, there were a lot of McIntoshes that went to Ulster before coming over here. That fits. Probably, (knowing my family like I do), there was some sort of insult, slight, argument, so other dispute that led to them leaving Scotland. Again, knowing my family like I do, I can see them leaving and not saying goodbye, and then blending in with the Irish to avoid the relatives back in Scotland… OR…
… either in Ireland or the U.S., they found it advantageous to throw in their lot with the Irish. Or, maybe both happened.
But, if we are truly related to John McIntosh, we came in, like your family, through Canada. McIntosh was born in New York, near where I lived. He got so annoyed with the new country (the U.S.) that he moved all the way to Upper Canada to get away from them. This certainly sounds like my family, doesn’t it? 😉
Oh, and I’m pretty sure my Granny was Presbyterian. She could not talk by the time I met her, but she liked to watch church on TV, but apparently hated the Pope. (I didn’t know the difference at the time… I was about 10).
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