Hint, not Indians. Or other cowboys. Although gunfights did happen, they were relatively rare.
The most common cause of death was falling from a horse. Riding helmets had not been invented (and sadly, many modern cowboys refuse to wear them out of a sense of tradition), so head injuries were common. So was being dragged by the stirrup, although good boots could prevent that.
Dying by falling from a horse would also have been a common fate of mounted warriors - the most common training accident would have been a simple fall. A knight's helmet was designed to protect him from blows to the head, so might have helped reduce the number of brain injuries - or not. And, of course, horse trainers would have faced this as their greatest risk (and to an extent still do).