It's funny that most make "country" singers wear cowboy hats. But probably few have ever been within a mile of a cow, or a horse. I can't stand most of them.
The genre is referred to as country, or cowboy, or western. But most all that music is "citified." Rarely is the name "hillbilly" applied. I love genuine hillbilly music. I love Bluegrass music, with guitar, mandolin, fiddle, and 5-string banjo. It's heavenly music.
I was raised in Tennessee on true hillbilly music. In my teens I once caught a Greyhound bus across East Tennessee by myself to attend the Grand Ole Opry in Nashville, live, in the original Ryman Auditorium. I saw Roy Acuff, Granpa Jones, Little Jimmy Dickens, Minnie Pearl, Bill Monroe, and may other originals. They are all icons in my mind. Some may have worn cowboy hats, but most wore overalls.
Some of the best old-time hymns ever sung were by hillbillies.