I used to have a more detailed and evidence-based theory which I now of course don't remember, but I think it was that he wrote the letter and intended to try to make the duel end without bloodshed by deloping - but for that to work, he'd have to clearly signal his intention and make sure Burr did the same. I think he was going to play it by ear, and only delope if he thought Burr would go along with it and do the same same.

But if he thought Burr was going to try to kill him no matter what, Hamilton would kill him, then go home and burn the letter. And if he got killed himself, well, moments of death and adrenaline are confusing and hard to recall. He'd have noticed that in the war. People would believe what they read over what they saw.

Hamilton's bullet lodged in a branch above Burr's head; too far for it to be carefully aimed to kill, but too close (and in his direction, which it shouldn't have been) to actually have been fired into the sky.

I think Burr had to have shot first and instantly, either while Hamilton was taking aim or while Hamilton was waiting to see what Burr was doing before he decided. Then he was hit, and fired on reflex with his gun pointed in Burr's general direction. Hamilon certainly didn't clearly indicate his intention to elope by pointing his gun to the sky before or during the countdown, or his bullet would have gone somewhere else. And if that was definitely his plan, he should have started sooner, or h would have known that it would look to Burr like he was planning to shoot him.

I think that historically speaking, Hamilton waited for it (too long) and Burr assumed that meant Hamilton was going to try to kill him and took his best shot.