I do like the way that the linked "Daily Kos" piece bit on Peterson is basically entirely a "so what you're saying is..." directly stating he generally doesn't use the term.
I think my earlier characterisation is holding up.
Right-wing pop philosopher Jordan Peterson, the Toronto-based lecturer whose book 12 Rules for Life is a nonfiction bestseller, has built much of his reputation as an enemy of multiculturalism and “cultural Marxism,” though he rarely uses the latter phrase, other than in mostly secondary contexts. Peterson prefers instead to attack “postmodernism” as the final outcome of misbegotten academic thinking: “Postmodernism, in many ways, especially as it’s played out politically, is the new skin that the old Marxism now inhabits,” he says in a video titled “Postmodernism and Cultural Marxism.”
In 12 Rules for Life, Peterson explicitly links postmodernism with the Frankfurt School. He also shared the Daily Caller piece describing cultural Marxism as killing America on his Facebook page.
I don't really see what the "conspiracy theory"/"hoax" is supposed to be. Clearly wokism/postmodernism/critical social justice/cultural marxism/whatever-you-want-to-call-it is totally getting out of control, particularly in the US. What exactly are you disputing?
Just the name "cultural marxism"? It's not a very much used term, as far as I can tell, and as the article says, Peterson doesn't use it.
Some background reading here: newdiscourses.com/tftw-cultural-marxism/
So from what I can gather, there is an anti-semitic conspiracy theory on the right based around "cultural marxism", but the same term is also sometime used more accurately to describe a real phenomenon, which Peterson talks about without using the term "cultural marxism", which makes him an anti-semitic conspiracy theorist. 