Did he jump, was he pushed?

Of course he was pushed... Well, maybe it was a bit of both.  

Barr's resignation was a RINO, (resignation in name only). He was saved from the humiliation that befell so many of Trump's other acolytes who weren't quite subservient enough by penning a sickeningly sycophantic resignation letter.     

His semi-voluntary ouster was an inevitable consequence of his decision to state publicly that he had seen no evidence of widespread voter fraud. That undermined and infuriated his boss, who certainly would have canned him had he not preemptively resigned; but the writing was on the wall.  

So why did Barr choose this particular moment to stop sucking up to Trump? Probably he realized that life after Trump might be irrevocably irredeemable if he stuck it out to the bitter end, associating himself with the fanatics who were still pretending that the voter fraud in the election was a thing. He can now, after all, say that he left the administration, not the other way round; that might help wash off some of the taint.    

Perhaps he realized it was time to retire, write the definitive tell-all book, and get some of that lovely consulting money the revolving door entitles him to.