"Cultural Marxism" was coined by American sociology professor Trent Schroyer in 1973.
http://www.iep.utm.edu/frankfur/
As for "Cultural Marxism" the Frankfurt school used the term "Cultural Hegemony" and "the Culture Industries"
https://www.marxists.org/reference/archive/adorno/1944/culture-industry.htm
Lukacs and Gramsci both argued that the reason that the Marxist class theory didn’t play out was because Western values were too deeply entrenched, notably the emphasis on the individual over the collective, and Christianity. These had to be destroyed in order for the communist utopia to be achieved.
In 1919, Lukacs was declared the “Minister of Culture” in the short lived Hungarian Soviet Republic. As someone in charge of education, he introduced some of the first sex education classes, designed to undermine the traditional sexual views of the West which he believed was a step closer to accomplishing his goal.
Gramsci worked on his “prison notebooks”, which outlined his chief insight which is called “cultural hegemony”.
https://plato.stanford.edu/entries/lukacs/
http://www-personal.umich.edu/~hfox/gramsci.html
Cultural hegemony stated the various factors that made Western civilization what it was, and was thus preventing the communist revolution. The key was to continue the “class war” but at the same, undermine these factors from within. These factors were:
Christianity
Authority
Sexual restraint
Personal responsibility
Heredity
Law
Truth
Family
Patriotism and national unity
Community
Conservatism
Language
Tradition
Gramsci believed that the media and academia had to issue a “counter-hegemony” message, which would help undermine these planks of Western culture.
https://onlinelibrary.wiley.com/doi/full/10.1002/9781118430873.est0614