Texas has mostly cows and cowboys who live at home. Montana and Wyoming have open range and cowboys ride out to do roundups seasonally for calving, branding, ...
My dad was a cowboy in Wyoming in the 60s. He had stories about little cabin shacks with pots of coffee and stew that just cooked constantly.
I'm not American but doesn't Montana have all these kinds of ranch-ey things? I'd argue that's more classic, hard work, cattle farming and less driving a Ford F150 through suburban Houston and yelling 'yeehaw' while explaining you were too tired to go to work at some fast food chain.
But that's pretty much all stereotypes that appear oversees.
I'm an east coaster, so basically the anti-cowboy, but Montana is another one that does come to mind. In King of the Hill, they go to Montana to do the biggest actual cattle related thing of the show.
Alberta, Canada also stands out to me as "cattle county" and they are Montana's neighbor. But as you said though, I'm not sure if that culture is the same as Texas style cowboy culture. That's also why I was interested if anyone outside the US would see their cattle folk as being more "real" cowboy than they see America's cowboys.
There is a saying "all that, no cattle" to refer to someone who dresses the part but has no experience, more of a cowboy role playing, and we definitely have a lot of that everywhere in America.
I don't think they'll top Texas, but I've heard that cowboy culture is popular in Japan. Probably the same way weebs are over here, uber-popular among a small group of die-hard fanatics.
Australia definitely has some of the same culture in spirit, if not in hats and boots. Steve Irwin would've made a good cowboy.