But that's not what people think of. When you say "I'm Irish" thoughts jump to leprechauns. That is not my culture, and therefore I should not be defined by it.
Nor is it an Irishman's culture. When I think of Ireland I think of a modern, predominantly Anglophone (for better or worse) very European country with a vibrant cultural scene. I think of a place to do business, I think of a place where I can see some pretty interesting Neolithic sites. When people think of Scotland, people think of guys in kilts, alcohol and Nessie. I know I actually wear kilts, drink whisky, speak Gaelic etc but that is mostly just a stereotype. Why would you let stereotypes get in the way? Surely the "Celtic"/Gaelic stuff is just one side to being Irish?
Is it perhaps the case that
your thoughts are the ones that go to leprechauns and that this is in some way related to a sense of very mild prejudice and cultural cringe against your fellow Irishmen and the concept of Irishness? You say that there is nothing wrong with being Irish, yet your cultural cringe is very obvious.
And people do call Canadians American, but they do it for a laugh. A joke, if you will, to get their hackles raised. But they know the difference. They understand that what they see as American is different from Canadian. They understand the culture changes.
A very good friend of mine is from Brazil and he told me once that he was annoyed that the USA had "hijacked" the term "American" and that he was proud to be an American too.
If everyone in North America were branded as American, no nod is given to a separate country and nation.
But according to the majority of Unionists, Northern Ireland is not a separate country or a nation, it is a province. Most Loyalists define it expressly as that and get offended if you say that it is a nation because they maintain it is a province of "Britain".
I see no reason why I should have myself defined, even if it is technically correct, as something I am not. Northern Irish is the equivalent of saying Canadian instead of American; preserves my difference and sense of self.
But it is quite possible to describe yourself as both Irish and Northern Irish in the same way that people from Northern England describe themselves as Northerners. Northern England is roughly as culturally distinct from Southern England as Northern Ireland is from the rest of Ireland. I don't see what the problem is.