The word censor comes from a Roman office of the same name, and so censorship has long had the meaning of official suppression of expression. This is particularly true in the US, where the First Amendment prohibits the state from exercising its power to block expression. The question arises about whether official power must mean official government power, and the answer is no. It has also come to include private, monopolistic economic power. For instance, in 1993, Wal-Mart instituted a policy of not carrying "objectionable" material, and given the company's market clout, when they refused to carry the original album In Utero by Nirvana, the band changed its cover art (no fetuses) and a song title (Rape Me became Waif Me). The policy has been widely decried as censorship, based on Wal-Mart's economic power and the resulting "artistic" suppression (scare quotes here because this is, after all, Nirvana).
It is a matter of opinion where to draw the line for applying the word censoship. There's no agreed-upon measure of power or size (in the case of organizations) at which private action becomes censorship.
The term self-censorship goes back to at least 1845, where we find this from Dashes at Life with a Free Pencil by Nathaniel Parker Willis
One year of such united self-censorship would so purify the public
habit of news-reading, that an offence against propriety would at
least startle and alarm the public sense;
The subject is the (ostensibly free) press ostracizing an individual on its own accord. This is still suppression by a collective. The OED credits Freud (Collected Papers, Volume IV, 1925) with identifying a personal censor
Dream-formation takes place under the sway of a censorship which compels distortion of the dream-thoughts.
Of course, Freud's censor is unconscious, while now we include deliberate censorship through fear of the consequences of free expression. Consider this from Queer Judgments: Homosexuality, Expression, and the Courts in Canada by Bruce MacDougall:
Self-censorship flows from that other constraint of expression, censoriousness. Censoriousness is one of the tools used to try to
accomplish censorship, often self-censorship, by others. It can
take the form of direct criticism but it might instead occur in the
form of ridicule, disparagement, or trivialization.
Note that the derived word censorious has always held the meaning of severely-critical fault finding, independent of official sources.