In the traditional Western ranching culture hat etiquette has always been around. As we have become more mobile it has become less so but it is still considered good manners in many places. When you enter someone's home, knock on a door, or speak to an older female (someone's mom/folks), taking your hat off is a sign of respect. This is where the term "Hat in Hand" actually means something.
Many people who own horses but may not necessarily be in the out-west ranching community may not be as in tune with it, but if you go to a place like Jordan Valley Oregon or a ranch in very rural Northern Nevada/Southern Idaho/Owyhee County ID to one of the ranches there, it is more normal, even if half the people are wearing ball caps these days.
I grew up somewhat in the Western ranching culture, going back to my paternal grandparents homesteading in central Montana, when a homestead was a half-section instead of a quarter. They didn't know each other until they took up adjoining 320s alongside one of the few year-round (but small) streams in the area, but after getting married had an entire square mile, which they eventually proved up, unlike most homesteaders.
My first full-time job after high-school was as a ranch-hand, long before most cowboys started wearing ball-caps. And yes, it was indeed traditional good manners to take hats off indoors, or in the presence of women. Some guys even took them off (as well as work-gloves) when meeting some other guy for the first time, and shaking hands.
Remember that starting to change in the 1990s, especially when (as somebody noted around then) that "most working cowboys wear ball caps and running shoes anymore. The Californians who move here wear big hats and 'cowboy boots,' especially when they go to a trendy restaurant."
In fact, my grandmother (who homestead by herself in 1919) always wore a dress, even when she went hunting--and she was an avid hunter and the best shot in the family. She regularly used a .22 pump to wingshoot gamebirds--which may or may not be due to knowing Ed McGivern, who also lived in Lewistown. She eventually became superintendent of schools in the county, and had Ed give shooting demonstrations at the high school. It was definitely a different time....
