Thanks for the history lesson. Very well put, but I would bet some confusion still exists!
There was probably misunderstanding in my post, because I do not use "gaited" horse & "racking" horse as the same. The term "Racking" horse I use is a registered horse that must gait.
So, yes, some gaited horses do trot & gait, some Racking horses gait & trot, but usually it means they are mixed, as you say, and the usual reason they mix breed is for color.
The Racking horse, around here anyway, is almost always black & the breeders will sell off any that trot & not use them for breeding. Again, they are only mix-breeding them for color, such as the Spotted Saddle horse. Therefore, they maybe show more "pure" horses without the ability to trot.
There are lots of gaited horses in the south, and the breeders do not want to keep any that trot. So in other places you may see a Registered Racking horse trot, but most physically cannot do it.
The ones that are mixed & can trot & gait, the breeders tell me that if it gaits after a walk, it is mixed breed TW or Racking. If it gaits after the trot, it is mixed with Saddlebred.
As far as other countries, that I do not know!