I believe the ones you see players using in those old pictures are using the kind that are skinny at both ends, and have a small sliding shoulder pad on them.
Also, my strap (levy's) is padded on the wide part, but not on the skinny part.
The Lone Ranger and Tonto were ridin' down the line
Fixin' everybody's troubles - Everybody's 'cept mine
Someone must have told them I was doin' fine...