A lot of horses on youtube don't really do the gait they're "suppose" to. Search for a TWH and you'll see more rackers and step pacers than horses doing a true running walk. You search for fox trotters, you'll probably find the same. Lots of horses within that breed that don't actually fox trot. I don't have a good eye for fox trotters, so I can't find you a good "ur example" video.
Fox trotters also have a walking horse like gait -- the flat walk. It's basically a slow running walk. Heck, I know several fox trotters who do a running walk instead of a fox trot. Horses don't read their papers; they don't know what they're suppose to do.
Also, you have 10 trotting horses. You have them trot out in front of you. You have 10 different quality trots. Same with gaited horses. They may do the same easy gait, but have a different way of performing it. It'll look different.
There can be some play within the gaits. Some horses perform a fox trot with more lateral timing than others; some perform the same gait with more diagonal timing.
Basically... It's confusing. You have to develop an eye for it.