The circumstance of Carlson's departure - no final week or month of shows - indicates that, regardless of the PR blah-blah, either Carlson was fired or Fox was imposing restrictions on its opinion show staff that Carlson refused to work within. Either way, Carlson will resurface in another or in his own new venue. Like throwing water on a big gasoline fire, this will probably just spread Carlson's influence, not silence him.
The circumstance of Lemon's departure - if one believes Lemon's claim - is that, despite similar PR blah-blah, Clown sCews Network manglement notified Lemon's agent rather than speaking to him directly. Cold, and probably indicative of him having been canned rather than an amicable parting of ways. Maybe Lemon should have figured out that venting his misogyny was only safe if his sole targets were conservative and R women. He may resurface, but IF he does, I doubt he'll be as successful at it as Carlson is likely to be.