That's the story Hamilton's seconds told. Burr's seconds said differently.
Don’t the winners always get to write the history? :-)
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.