I thought in other places/times of history when duels were common, it was tradition that the challenged party could choose the weapon. But in the Western novels I've read, this never seems to happen, the person starting the dispute just seems to be able to decide himself whether it is going to be settled by fists, knives, guns, etc. and the person challenged seems to have to go along with it.
Ritualized dueling had become extremely rare by the "wild west" period of the 1870s onward and many states had been enacting anti-dueling laws back east. Likewise, there's little to no evidence of the "showdown at high noon" actually taking place. Most conflicts where someone wanted to kill the other person would just involve shooting or stabbing them.
The closest thing to a "duel" in the west (and likely inspiration for this trope) was when Wild Bill Hickok killed Davis Tutt in 1865 in Springfield, MO. During a card game, Tutt claimed Hickok owed him more money from a previous game and took his golden watch as collateral. Hickok was outnumbered but warned him not to wear the watch in public or he'd shoot him.
Tutt brazenly wore the watch the next day and word reached Hickok quickly. He showed up with his pistol cocked and in his holster and warned Tutt "Don't you come across here with that watch." From a distance of about 75 yards both men stared at each other, then drew their guns. Tutt was first, Hickok was more accurate. Both fired simultaneously, with Tutt missing and Hickok hitting him in the ribs. The judge allowed the jury to apply the unwritten law of a "fair fight" and he was acquitted of manslaughter.