True. But that last bit with Murray was when he fully embraced *every* aspect of being "Joker." Full costume, full makeup, and totally relishing the mayhem. And the transition from Arthur to Joker actually correlates pretty well with his appearances on the Murray show.
First appearance is all imagined in his head (and the imaginary elements in the film only seem to happen when he's still Arthur, not when he's Joker IIRC). In this imagined appearance, Arthur is embraced (even literally) for wanting to put smiles on people's faces. It's 100% Arthur, but he's living a lie.
Second appearance is the footage from the stand-up routine. He expected to be embraced here too (bringing smiles), but instead he was mocked and ridiculed. This appearance coincided with key points in the film that lead Arthur to realize that his life has been a deception/delusion. Still Arthur here, but starting to change.
The third appearance is obviously full Joker. Going off his meds and killing Randall was the official shift, but shooting Murray was when he had fully embraced who he really was. Instead of killing himself on the show, he effectively only killed "Arthur" (and obviously Murray
). All Joker going forward.