I'm reposting a larger piece I wrote as it does, in part, touch on Burr's own remarks:

Its been 214 years since July 11, 1804, when Aaron Burr shot down Alexander Hamilton in a duel at Weehawken, Hamilton passing away from his wounds the next day. The duel is an enduring piece of American historical memory, and a source of controversy, perhaps none more notable than the conduct of the duel itself, and how it proceeded. Excluding Burr himself, who we will return to however, there were only two witnesses to the duel, the seconds Van Ness and Pendleton, men who never found themselves in agreement on key points of order of events. Contrary to popular account, they viewed the encounter, rather than turning their backs at the moment of firing, but remembered things very differently. Other men present, Dr. Hosack and the boatmen who had rowed everyone across from New York, as was expected, did remain a discrete distance away as to not be witnesses.

In preparation for the duel, the arrangement had been made as follows, a fairly standard procedure:

The parties having taken their positions one of the seconds to be determined by lot (after having ascertained that both parties are ready) shall loudly and distinctly give the word "present" - If one of the parties fires, and the other hath not fired, the opposite second shall say one, two, three, fire, and he shall then fire or lose his shot. A snap or flash is a fire.

The ground was staked out on a north-south axis, with Hamilton winning the right to choose his position, taking the northern side, a curious choice in Chernow's estimation:

Because of the way the ledge was angled, this meant that Hamilton would face not just the river and the distant city but the morning sunlight. As Burr faced Hamilton, he would have the advantage of peering deep into a shaded area, with his opponent clearly visible under overhanging heights.

Alternative arguments have been made that Hamilton believed the angle of the light would better illuminate Burr for him, however. The only direct commentary we have came when he put on his glasses, noting "In certain states of the light one requires glasses", a comment that detractors took to be ominous, and how one understands his decision of position - the better or worse position in his personal estimation - is severely impacted by how one views his motivations sketched out below.

After taking their places, what can be said with absolute certainty is that two shots were fired, and one man was mortally wounded, but little more will ever be known with 100 percent certainty between Pendleton's command of "present" and Hamilton lying wounded on the ground.

After the duel had occurred, as was common when the encounter gained such public notice, the two seconds released a joint statement but disagreed on the most crucial part of who fired first:

And asked if they were prepared, being answered in the affirmative he gave the word present as had been agreed on, and both of the parties took aim & fired in succession. The intervening time is not expressed as the seconds do not precisely agree on that point. The pistols were discharged within a few seconds of each other and the fire of Col: Burr took effect.

Once the controversy began to boil, they released competing addendum, each in favor of their own Principal. Pendleton's statement established that Hamilton had confided in him the intention to reserve his fire, which was also expressed in the prepared remarks Hamilton had written prior, and that he had reiterated this just prior to the exchange when he mentioned that he had not set the hair-trigger "this time". Several others claimed to have heard similar communications, and additionally, Pendleton noted that afterwards, in the presence of witnesses, Hamilton lamented "Pendleton knows I did not mean to fire at Col. Burr the first time" and also seemed to be unaware his pistol had fired, warning the boatsman handling it that it was loaded - also expressed by Dr. Hosack in a letter to William Coleman several days later. Pendleton asserted that Hamilton had only fired after being hit, an involuntary reaction which sent his bullet high above and too the side of Burr, which he backed up by claiming to have returned to Weehawken and recovered a branch from that spot with a bullet hole in it.

In their ensuing duel of the pen, Van Ness gave his own version, first noting that Hamilton had shown no reluctance prior, and in fact practiced sighting the gun, and then donned his aforementioned spectacles to try again, which could only have been a demonstration of intent. He then described the sequence as Hamilton firing, and Burr waiting some five to six second to return fire, in order to let Hamilton's smoke dissipate.

It is of some interest that in later accounts, Van Ness changed this, making the interim smaller. In having Burr fire first in his own account, Pendleton absolves himself of responsibility, but in Van Ness's version, Pendleton would have been very much to blame, as it would have been his duty under the rules of the duel to count off Burr's three second window. The change by Van Ness may have been simply because he reevaluated his recollection and was less certain of the interval, or it my have been a conscious choice to avoid unnecessary imputation of Pendleton's own honor, an act which could have potentially provoked its own duel.

In any case, as least putting aside the precise interval, Van Ness was sure of what he had seen, since:

On this point the second of Col Burr has full & perfect reccollection, he noticed particularly the discharge of G H's pistol, & looked to his principal to ascertain whether he was hurt, he then clearly saw Col Bs pistol discharged. At the moment of looking at Col Borr the discharge of G H's pistol he perceived a slight motion in his person, which induced the idea of his being struck, on this point he conversed with his principal on their return, who ascribed that circumstance to a small stone under his foot, & observed that the smoke of G Hs pistol obscured him for a moment previous to his firing.

Those are the only eyewitness records we have, as, again, the boatmen and the doctor, to ensure the veneer of deniability, did not observe the exchange. On the whole, the version favorable to Hamilton is generally favored, even if Burr has his defenders. The statements of intent that Hamilton made expressing a desire to reserve his fire for the first exchange and those in the boat after the duel are corroborated, insofar as possible. Burr has his supporters, some who would go so far as to believe Hamilton maliciously planned all of that as a backup plan, to ensure that if he did die, he would at least have destroyed Burr as well, but there is no real proof of this, except for Burr himself.