I don't think it's a no true scotsman thing, although as I was writing the comment I did worry that I was veering into that territory.

How one defines a Democrat varies, and only some ways of defining it make sense with the sort of Democrats you describe, and I don't think the overlap of "Democrats" and "people who identify with a specific political -ism" contains those. This will vary a bit depending on how one is interpreting "identify with a specific political -ism". I think this is where the disagreement lies.

You confuse party affilation with political beliefs. It's a mistake that comes from living in a two party state and having a media that tries to convince everyone that red and blue are the two political beliefs that one can have.

It's very worthile to have a mental concept of political beliefs that goes beyond party affilation.

George Orwell would say that the media tainted the language in a way that makes it impossible to analyse political beliefs in your vocabulary.