Jos. Ellis’ book Founding Brothers has a rather thorough discussion of the duel, but I can’t remember all the details. I think it concludes that Hamilton’s first shot went way over Burr’s head, breaking a tree branch, but Hamilton did fire twice. It also concludes that Burr may have been trying to wound H. in a non-lethal area. One positive effect of the duel is that it destroyed any remaining respect for Burr, who seems to have been an unprincipled political manipulator.
I've a couple of bios of both men and quite naturally the Hamiltonians stick with his story and the Burrites stick to theirs. We'll never know, but we do know that Hamilton knew he was baiting Burr and that he knew Burr was a dangerous man who would defend his honor.