This dear reader does believe he was lucky to be alive.

On that same site you just linked I found the Code Duello that was written in 1777. Perhaps this is the code they followed.
By the rules written, it seems it was very dangerous business indeed.
Here's
Rule 13. No dumb shooting or firing in the air is admissible in any case. The challenger ought not to have challenged without receiving offense; and the challenged ought, if he gave offense, to have made an apology before he came on the ground; therefore, children's play must be dishonorable on one side or the other, and is accordingly prohibited.

So by this rule deloping( a word?) would be childish, right? So when the fight begins, they are meant to try at least injure.

If I understand the rules correctly, if Willoughby had apologized before taking to ground (not sure what that means) the duel could have been avoided. Does that mean that he must apologize before the meeting at the time and place, or when they meet he could have apologized before the fighting started? It said too that the challenged chooses the ground, and Challenger chooses the distance.

I'll link the Code Duello below, and I found rules 8, 10, and 11 quite apt for our story here.

I would also be inclined to think pistols, because it would be harder to imagine both of them returning uninjured with swords. ?