I miss my Mustang mare Desert Rose. She was gaited though I never figured out exactly what lateral gait it was. It was 4 beat so it wasn't a pace but it was like I had extensions on my legs and was walking with really big steps. You let you legs swing like you are walking just behind her shoulders as the went forward and back. One day we had some dogs rush us on the road. She hates dogs. She took me up an embankment that I would have had to grab the sapling trees to pull myself up if I was walking. One of the dogs had been up on the embankment and there was NO WAY she was going to let him keep the high ground on her. I just gave her the rein and flattened myself to her neck.
Could have been a full out fast singlefoot AKA rack or a running walk. Fox trot is diagonal.
What people generally don't realize is that ranchers used to turn out the kind of horses they wanted into the broomtail herds, and would should the broomtail stallions and put out a Morgan, or a Standardbred, or a pudding foot, or a Quarter Horse stud to get the ball rolling.
In parts of the western mountains, especially in eastern Oregon, you still see the occasional "pudding foot." A big, strong, rangy horse with some definite draft ancestry visible in the feathers and legginess, and a strong tendency to being rawboned. They are tough, and if I could find one, I'd buy it.