Do Scots, Welsh or Irishmen hate it when Americans call themselves Scots, Welsh and Irishmen?

I met a scottish lady a few years back and explained I was Scottish. During the conversation I told her my family left Scotland (damn Campbells.. but that's another story) around 1660 and settled in Belfast before leaving around 1720 for America. So she says "Oh no Deary, you're what we call Scots Irish (as in not really Scottish)". Why? Because they stayed in Ireland for 50 years without intermarrying? Big deal. I do have some Irish blood but it comes from a completely different line. I don't really care but I think a lot of Europeans are picky about their heritage and culture. There are far more Irish in the NE US than in Ireland, and far more Scottish in the SE US than in Scotland.