Right. But that's never what the discussion is about. This is the first time I've seen someone say, "it's not just a next-token predictor because a bunch of the training isn't about predicting the next token." I have seen many, many people say "it's not just a next-token predictor because there's all this smart stuff going on inside the model." Even here, in the comments for an article that's all about the former, most of the objections to "next-token predictor" are of the latter form.
Yup, you are mostly right.
I guess the people in my camp find the "it's just a next token predictor" stupid in that it's like saying "it's just a bunch of carbon and hydrogen", but it's also one of those things where people like to think they are clever because they think they are theoretically correct. But they aren't even that. So it's like double stupid. But the "next token predictor" part is at least technically correct (like, carbon and hydrogen right) for pretraining, so the debate can't really be won there.