Absolutely right - enough of them were Indians to make that less likely. By the time of the cattle drives the extinction that had begun with Cortez and the dislocation that was largely a result of the British model of cultivation colonization were largely over. And the cattle drives themselves didn't last long after the railroads came through, and that lonely range cowboy playing his guitar to the cattle on the lone prairie (ever wonder where these guys carried guitars? I'm thinking it must have been in the camera van) morphed into the working stockman who survives to this day.
Now, "cowboy" as a sobriquet conjures (to me, at least) the image of a guy riding at full gallop into town, sixgun firing wildly into the air. This, too, is a bit of a stretch inasmuch as the townspeople tended to be considerably better-armed than the cowboys and took a dim view of unwashed trail jockeys shooting up their tiffin. Coffeeville and Northfield come to mind. Home team 21, Cowboys 0.
When Euros accuse Bush of being a cowboy, they confuse a cowboy with a gunslinger (on occasion cowboys qualified for both roles) from old western movies. The concept is strictly Hollywood, and has nothing to do with the actual old west.
Euros can't be taken too much to task, though, since Americans frequently use the term in the same way.
Brings to my mind a guy working for $10/month and found for a small rancher. His job is fixing fences, rounding up strays, breaking horses and doing periodic round-ups while trying to stay out of the sight of the rancher or foreman.
The Cowboy was Black, White, Mexican or part Indian, and usually crippled to some degree by 30; an individualist with a sense of right and wrong as cowboys that tried to cut corners or stray from the path usually ended up dead.
I consider the term Cowboy to be a title of Honor.