I see your point, but let me clarify why I say that. I say "I am Scottish" in America, because everybody knows that I'm American, they want to know if I have Scottish ancestry. In the US, it's just easier to say "I am Scottish" than "I have ancestors who were Scottish". If I ask somebody "if they're Scottish", I know they're American, I want to know if they have Scottish heritage.

Now, when I'm abroad, I clarify like that, because they don't already know that I am American. Like when I'm in Ireland, people will ask me if I'm Irish and I reply "I'm American, but have Irish ancestry".