That's not the definition of censorship. Censorship is the suppression of information by any authority who has the ability to suppress said information. The definition doesn't require a government or any aspect of coercion.
If I run an online forum and I start removing posts I don't like, that is censorship. I'm allowed to remove those posts because it's my website, but it's still censorship.
If I choose to not throw fuel on a fire, I'm not suppressing the fire. Similarly if I choose to not share gossip about something for which I have no first hand knowledge I am not suppressing it. Hence no censorship.
If I run an online forum and I start removing posts I don't like, that is censorship. I'm allowed to remove those posts because it's my website, but it's still censorship.