Patti,
Originally it was just the Canadian River with tributaries of Deep Fork, North Fork, and South Fork. North Fork of the Canadian became know as the North Canadian which, I ass/u/me, led to the Canadian being called the South Canadian. The South Fork of the Canadian became known by the name of its major tributary, Gaines Creek.

I bring this up to caution others in reading the ORs where South Canadian sometimes means "South Fork of the Canadian" and "Canadian" means what 'local residents' call the South Canadian.

So what do you call the river from Eufaula Dam to the Arkansas? Mixed Canadian? LOL