I doubt we'll ever get everybody to agree on what counts as "censorship"; but, I don't think we need to either. Rather than arguing about what we think "censorship" ought to mean, it would do us good to pay more attention to how the word is actually used.
By law and government, many different kinds of speech/communication are suppressed and/or restricted–fraud, perjury, defamation, violent threats, conspiring to commit crimes, impersonating law enforcement or other government officials, unauthorised publication of confidential information (medical records, intimate images, nuclear launch codes), inciting imminent lawless action, practicing regulated professions (such as law or medicine) without the appropriate qualifications and licenses, the proverbial "shouting fire in a crowded theatre", encouraging vulnerable teenagers to commit suicide, etc, etc, etc. The vast majority of people agree with most (or even all) of these in principle, even though there are still disputes about them at the margins–indeed, most people would agree that the suppression of many of these kinds of speech is essential for society to function. Yet very few would call suppressing these kinds of speech "censorship"; why? Well, the word "censorship" is largely a pejorative term–you'll only call it "censorship" if you disagree with it. If you believe certain kinds of speech ought to be suppressed–as almost everybody does–by that very fact it can't be "censorship", that's the way most people use the word. Now, there probably do exist some people who will say "Yes, banning X is censorship, but I support it, that's the good kind of censorship"–but, those people are very much a minority, most people don't use the word that way.
Since we'll never be able to fully agree on what kinds of speech are justifiably suppressed, we'll never be able to fully agree on what counts as "censorship" either. However, I think a lot of these debates get derailed into pointless semantic squabbling, by people who appear unaware of the "it can only be censorship if you disagree with it" principle which governs how the majority of people use the word. Once you are aware of that fact, it becomes clear that debating what forms of speech suppression count as "censorship" is a distraction from the real issue – debating where to draw the line between the good cases of speech suppression and the bad cases of speech suppression.
Another issue underlying many of these definitional debates, is different views on the significance of the public-private distinction, the distinction between governments and private corporations. To some people, that distinction has immense ideological, philosophical, even moral significance – they believe that governments and private corporations are fundamentally different kinds of entities, and very different standards should be used in judging them. Others think that governments and large corporations have many similarities; that even if there are some fundamental differences between them in theory (although even that is disputed), those theoretical differences have far less relevance in practice; that they should be judged by fundamentally the same standards – to these people, what matters is what is being done, not whether the public or private sectors are doing it. The first school of thought is naturally attracted to government-centric definitions of "censorship", such that "censorship by the private sector" becomes definitionally impossible; the second school of thought is instead attracted to definitions of "censorship" which are neutral to the government-vs-corporations distinction, so "censorship by the private sector" is entirely conceivable.
I think, once again, too much time is wasted by approaching this as a debate over the correct definition of a word, rather than focusing on the underlying philosophical differences. Inevitably, people with radically different philosophical starting points are almost inevitably going to adopt different definitions of key words, and it is impossible to answer "whose definition is right" without first answering "whose philosophy is right". Since philosophical disputes are notoriously unresolvable, we are never going to agree on the definitions either, so we might as well just all stop arguing about the later. At least the philosophical debates, even if unresolvable, have some profound significance, for how we ought to live our lives and organise our societies, for very Truth and Goodness themselves; the same cannot be said for debates over how to define a word.