still have two 90 degree angles and two 270 degree angles

But you just said internal vs. external angles was a distraction that people shouldn't get hung up on.

To be clear, apart from that one bit, I agree with you completely. I don't even necessarily disagree with what you said back there per se. I just don't think it was useful to bring it up because even if it isn't explicitly in the definition of a square, it is an implicit assumption that when talking about the angles of a shape, you're always talking about all internal angles (or, equivalently, all external angles, just no mixing), so getting "hung up on" internal vs. external angles is not a bad thing.