I have a few corrections to my last post:
1. According to Harry Carrey Jr. the horse that I called Henry named Hank (or Pank or something like this). Which also proofs that the horses name ain't Banner.
2. I don't think Hank was owned by the Fat Jones Stables because in all of his movies he was along with horses from Hudkins Brother (including They Died with their Boots on and a lot of Republic westerns(Roy Rogers, Red Ryder & co). Petrine Day Mitchum wrote that all horses from these movies were provided by Hudkins). I saw Hank with Fat Jones horses just in Rio Grande and Red River (in both he was ridden by JW).
3. He also rode Hank in War of the Wild Cats and The fighting Kentuckian
4. In She Wore a Yellow Ribbon he rode Banner in the whole movie besides the scene were he jumped over the fence at Sudros.
I also would like to mention that Banner was in a lot John Wayne westerns where he wasn't ridden by him, such as Tall in the Saddle, Red River and Rio Grande.
As well I have done some research about Dollar and Dollor.
John Wayne rode the same horse with a big blaze and 3 white socks in Chisum, The Undefeated, True Grit(before the fight) and the beginning of Rio Lobo (ofcourse in the most movies he rode more than one horse, for example there were 3 to 4 different sorrels he rode in Rio Lobo). Chuck Roberson wrote about Rio Lobo "[...]Duke's horse Dollar[...]" (e.g. "The Fall Guy", page 283). So I'm pretty sure this horse called Dollar.
Then he has the sorrel with a more narrow blaze and two white socks at his back legs, which he rode in Big Jake and after Big Jake. I think this horse called Dollor. Also there is a Breyer modell from this horse called Dollor.
The most sources call the first horse Dollar and the second Dollor (, besides the book "1000 famous horses" who called the first Dollar and the second Old Dollar, another named the second as "Dollor or Old Dollar"). A few like the one @kkaterinabella_007 recalls, as well as Petrine Day Mitchum in her book "Hollywood Hofbeats" say the first horse called Dollor and the second Dollar.