Censorship has a very simple definition, and it can be conducted by private companies. From Wikipedia:
“Censorship is the suppression of speech, public communication, or other information, on the basis that such material is considered objectionable, harmful, sensitive, or "inconvenient." Censorship can be conducted by governments, private institutions, and other controlling bodies.”
“Real” censorship is just censorship, and that’s the definition.
The reason I used the word "real" is probably because of my environment (ex-communist european country). I was born after the fall of communism, but of course the environment has a big influence on the lens through which you see things (family, friends, stories, history classes, etc.)
So when I see what is called censorship in US, it seems a bit funny and strange to me. I immediately think how people in my country during communism would be happy if only that was called censorship and if they had a possibility to use alternative channels to exercise free speech. But of course, I accept that this might be my biased eurocentric view of things and that from a different cultural perspective "real" means something else. Perhaps it will be like that even in Europe in a few years, who knows. We don't have much problems for now because we all use US social media and they don't really react to foreign languages, except a few universal "trigger" words like antivax. At least for smaller languages and countries (central and eastern europe)
Present-day social media censorship in the US is not "just free market", and is markedly similar to the kind I experienced growing up in a now-ex-communist eastern European country in the 1970s-1980s. Outright bans were not politically tenable by then, but forcing the opposition to express their views on fringe "alternative" channels was a popular strategy [1].
WeChat does not let you share Winnie the Pooh. Nominally, Tencent is a privately owned company freely choosing who they provide a platform to [2]. But in reality, they have no choice but to ban Winnie the Pooh, unless they want a state apparatus to make their lives increasingly difficult. Just because something is censored by a private platform doesn't mean that it's the "free market" of ideas and not a state or the government wielding its power.
Large US companies are also part of an industrial–congressional complex, with lobbying and political contributions on one side, political approval and threats of regulation on the other. Large tech companies are deeply and inseparably intertwined with the state and the political parties (both of them), based on the granting of reciprocated privileges. They know full well that if they ban the wrong speech (or refuse to ban the "right" speech), they face being regulated out of existence. Indeed, seeing this threat, we see them scramble to align with the incoming administration.
The kind of social engineering that led to the present bans was very popular in communist Eastern Europe as well. Did you recite an anti-government poem at your barbecue? You might find yourself banned from your favorite pub permanently! Why? Nominally, it was the bartender exercising his right not to serve you: after all, he shouldn't have to suffer potentially vocal ("verbally aggressive") imperialists in the establishment he runs. But behind the nominal reason was plain, state-mandated censorship. They'd have risked a bunch of misfortunes by not banning you. These could range from relevant, like the next few beer shipments "mysteriously" getting damaged during delivery, to completely random gaslighting-esque punishments, such as the bartender's daughter not getting admitted into the local high school. A plethora of plausibly deniable, no-outright-ban social mechanisms existed to make life difficult for people who disagreed with the official narrative, and to encourage people to ostracize the disagreeable. It required only a few well-placed entryists, made sure that ordinary citizens had skin in the game, and was a lot easier to handwave away than the black cars and heavy-handed approach that riled up the opposition in the previous years [3].
“Censorship is the suppression of speech, public communication, or other information, on the basis that such material is considered objectionable, harmful, sensitive, or "inconvenient." Censorship can be conducted by governments, private institutions, and other controlling bodies.”
“Real” censorship is just censorship, and that’s the definition.