Profit is definitely one motivation for censorship, but there has been plenty of censorship in non-profit-centric situations. Communist countries and other government actors have been leaders in censorship, with no obvious profit motive.
IMO, we should use "content moderation" for the former and "censorship" for the latter, congruent with historical usage. But people who are against content moderation will claim I'm being biased, even though I view the whole debate as a bit of red herring that distracts from the real issues. So I'd settle for "private-sector censorship" and "government censorship".
But in any case it's almost always counter-productive to conflate the two, to the point that it's a logical fallacy which should be named.
> i am not so sure if the distinction still exists when every conversation online is covered by private-sector censorship
That's certainly fair, and I think a particularly prescient observation in favor of the conclusion that it's time to severely restrict legal protections for social media platforms.
This implies you don't think "content moderation" is a "real" problem, which suggests to me you are biased. Also, "moderation" to me suggests mere moderation (consistent with fair, stated guideline) versus moderation "if you say things they don't like".
This kind of fight happens all over the place (e.g. what counts as hate/rape/murder etc), but I feel "moderation" is fairly neutral sounding without the implication of injustice befitting the act of being deprived of a a voice on omnipresent and monopolising platforms. It's also clear to me that some publications use comments sections as a way to manipulate public opinion, so cherry-picking/censoring them is an exercise in PR if (and only if) it isn't clear that result are being manipulated.