I wasn't aware that the Frankfurt School thought of themselves as communists. This is news to me. Can you refer me to any writings by them where they talk about this? Thank you.
Ah, apologies. I thought you were asking in good faith instead of trying to score some semantic point. In that case, you describe Marx as a "communist thinker", which would make the Frankfurt School(who were Marxists) communists by the terms of your categorization.
https://www.marxists.org/subject/frankfurt-school/
And Horkheimer is of course famous for saying, "If you don't want to talk about capitalism then you had better keep quiet about fascism."