If I'm not mistaken, BERT is a classifier (enters text, outputs labels) so it is not a "Language model", as it cannot be used for text generation.

The abstract of the original BERT paper starts with these words: "We introduce a new language representation model called BERT, [...]" The paper itself contains the phrase "language model" 24 times.

It might not be considered a language model today, but it was certainly considered one when it was originally published. Or so it would seem to me. Maybe there is a semantic shift which happened here?