My understanding is that Country/Western isn't a type of music. It's a descriptor of radio stations that play types (note the plural) of music that includes Country, Western/Cowboy, Hillbilly, etc. but mainly was in contrast to stations that played more urban types such as Big Band, Pop, Jazz, Rhythm & Blues, etc. In other words, it was a catchall label for all that hick stuff.
Re the set list:
"Sing Me Back Home" was written and first performed by Merle Haggard. Most country music fans associate it with him. I didn't even know the Flying Burrito Brothers covered it.
"Old Man from the Mountain" is also by Merle Haggard.
"Milk Cow Blues" was written and first performed by (James) Kokomo Arnold, but is probably better known as being done by Elvis Presley and Willie Nelson (among many others). Likely any country/rockabilly performance has its inspiration in Bob Wills' recording of it.
For Certified Nitpickers, the song variously called "Ghost Riders," "Ghost Riders in the Sky," etc. is more accurately "Riders in the Sky." It has been done by everybody and his brother. The rendition by Vaughn Monroe and his orchestra (which I believe was the first recording) was a huge hit, and in my opinion stands head and shoulders above most others (note for example, the different phrasing used for the third verse, driving it to a climax). But however it's styled, it's a great song and always a crowd-pleaser.