The article cites Trent Schroyer as coming up with the term, Schroyer's thesis is still that the Frankfurt School were cultural Marxists because they cited Marx in developing Critical Theory.
Great, fun and good except that Marx's theory that dialectical materialism was the driver of history, is based on certain things being objectively true, like conflict being caused by material needs-according to Marx anyway. Critical theory which gave birth to postmodernism
flatly rejects objectivity in knowledge. The Frankfurt school could be reasonably said to be influenced by Marxism, or post Marxist but Marxist-it aint.
Western Marxism did indeed depart from Soviet Marxism b/c piles of corpses will turn off anybody but none of the people constructing the shibboleth of "Cultural Marxism" have shown that the people they called "Cultural Marxists" were in fact, you know, Marxist.
The real reason "Cultural Marxism" was embraced by paranoid nutzoids everywhere is that at the time "Marxist" was the trigger word that led conservatives to circle the wagons much like "Muslim" is now. At best both Marxists and some of the ideological descendants of the Frankfurt School are both involved in social justice movements for their own reasons.