Socialist is an umbrella term encompassing everything from social democracy (also known as democratic socialism), which seeks to compromise with capitalism, to communism, which seeks to overthrow capitalism and create an entirely new economic order. The terms social democracy and communism also vary in how radically they oppose or work within capitalism. For instance, the US Democratic Party has some slight social democratic tendencies towards wealth redistribution, but is still a pro-corporate/pro-business political party, whereas many social democratic European political parties are more pro-labour. Similarly, the Chinese Communist Party was more open towards capitalist reforms under Deng Xiaoping and Hu Jintao than it was under Mao (and than it seems to be under Xi).
Marxism is an economic school of thought that seeks to understand history and society through the lens of class struggle. Not all socialists are Marxists. In fact, these days, many socialists don’t see class struggle as an important factor for understanding society at all. Many socialists these days (although this has been going on since the 1960s and earlier) look at history and society through the lens of environmental and identity issues (such as race and gender).
I didn’t watch all of the video, but from what I saw, the professors aren’t Marxists. Or, at least, they don’t express any Marxist leanings in these videos. Some people might call them adherents of cultural Marxism, which is a right-wing conspiracy theory alleging that liberal elites (such as media owners and university professors) are projecting Marxist thought away from economics into social issues (such as race, gender and identity) in an attempt to undermine society and destroy all tradition. It’s basically the old anti-Semitic conspiracy theories about Jews corrupting society through Jazz music and skimpy clothing, but in a new non-Jewish guise (although some of the more fringe cultural Marxist conspiracy theorists do, indeed, blame it on “the Jews”).