As long as i get to be the frog
I can't (and won't try to) speak for the other two, but I'd find the behavior of the player whose character insulted the tyrant while I was negotiating with him to be literally infuriating. You're not only keeping my character from achieving the goal of the negotiation now, you're keeping my character from ever achieving that goal with this NPC. It has vibes of PvP to me--and I utterly detest PvP.
The problem is that it takes 2 to tango. I don't believe we can give the DM a pass for playing the NPC to their conception while condemning the player for playing his PC to his conception.
The player insulting the tyrant didn't cause the negotiation to end. The DM choosing to have the NPC react that way did. Was such a potentially realistic NPC reaction? YES! But it wasn't the only realistic NPC reaction.
IMO. A DM should strive to play an NPC such that one or two remarks by one of the players will not totally upend the other players fun (in this case if other players were having fun negotiating then the potentially realistic NPC reaction to the insult was the wrong potentially realistic NPC reaction to go with). IMO the NPC's reaction, which the DM controls, is the direct cause of the negotiating fun ending, so why not lay the blame on the DM where it appears to belong?