I'll have more tonight, but some initial thoughts:
I thought the Bulls made a mistake having Gar speak first. That is, if their intent was to cement Gar's role as the lead decision-maker of the organization.
Gar did a nice job of emphasizing that he was in charge, balancing the explanation of why the coaching change was necessary while simultaneously praising Vinny, and evading questions of parameters for the next coaching search.
Then Paxson took the stage, and he clearly has more of a presence than Gar. Started out with a tone of genuine embarrassment when discussing the confrontation with Vinny. But he then kept talking, emphasizing how the confrontation shouldn't effect overall perception of him, a weird aside on his 'scrappy' upbringing, mentioning (a slip-up?) that he'll be talking to FAs, and culminating in a defiant statement about him 'not going anywhere'.
It sounded more and more like he was truly the man in charge, not 'a consultant'. Such a delineation isn't as much for pacifying curious fans (though that'd be nice) as it is about the perception around the league. Do those types (the kind that has influence over FAs and coaching candidates) feel based on this press conference that Gar has any more power? The first half of the conference made it seem that way, but it seems like Pax's pride got the best of him again as he re-cemented himself as the top executive.