Perhaps not the fastest gunslinger, but pretty darned fast, quite accurate and one of the most ruthless gunfighters ever - John Wesley Hardin. Attributed with at least 41 killings. Spent some time in my neck of the woods, Jacksonville, Florida. Should have been one of the most notorious, but I suspect not having a memorable nickname kept him reletively obscure.
Hardin was fast in the sense of “quick on the trigger” (easily angered and prompted to shoot), but few of his killings were stand-up, face to face encounters between adversaries who were both aware that a gunfight was imminent. (In fact, several of his shooting were from ambush, several more were the result of him responding to the threat of a knife or club with gunfire, and a couple were the result of him hauling out his gun and firing at a person who did not even realize that he had any intention of fighting.)
Hardin was dangerous (and was quite famous at the time, as well as among those who read about the West, today), but there is no evidence that he was particularly a “fast draw.” (He was accurate.) Hardin was never obscure, although his actual life did not lend itself to romanticizing movies, so he is not well known among those whose only acquaintance with the West is TV or cinema.
*He was short and fat, and rode out of the West
With a Mogen David on his silver vest.
He was mean and nasty right clear through,
Which was kinda weird, 'cause he was yellow too.
They called him Irving.
Big Irving.
Big, short Irving.
Big, short, fat Irving.
The hundred and forty-second fastest gun in the West.*
Ah, beaten!
131 were faster than he,
But Irving was looking for 133…
Ahem. The Lawless Breed, starring Rock Hudson and (loosely) based on Hardin’s (self-aggrandizing) memoirs.
Darn tootin’
I was quicker on the draw 
Before his death, young Jimmy Brown was considered one of the fastest actors with a sixgun draw. (no buzzer or go siginal, just stand there and someone moves first.)
YMMV
Try…
141 were faster than he,
But Irving was looking for 143…
Zev Steinhardt
Sharon Stone. (my idea of funny - I don’t remember her character’s name in the Quick and The Dead)
Well, there is always Luke Short. Short was not known as a “gunslinger” per se, but he was definitely good with a firearm and was involved in two seperate, well-documented gunfights. In the first one, he shot and killed one Charlie Storms:
In the second, Short killed one Jim Courtright:
Nobody is saying that nobody ever killed others with guns in the old west. The question was whether there were classic street duels with guns.
A guy who pulls a gun in a saloon on someone so close that he set his shirt on fire is a simple murderer. We know they existed. There were dozens if not hundreds, although not nearly as many as Hollywood would have us think.
Pulling a gun and killing someone sitting next to him doesn’t make Luke Short a gunfighter any more than Al Capone’s pulling out a bat and beating in somebody’s brains makes him a baseball player.