“Training” medals would be hard to define because there are a whole lot of gray areas.
Obviously, if you started your horse as a baby and are the only person that has been on him, you can be certain that you trained him through whatever level it is he’s competing at.
But what if someone buys a horse that was previously ridden by a professional at First Level, then goes on to successfully compete through Third Level? Did they train that horse? What if, instead of a professional, the previous rider that did First Level was a badly-riding amateur? What if the horse has never shown in dressage with anyone else, but he’s shown extensively under someone else as a hunter or jumper?
What if they had a colt starter put 30 days on the horse and get him going under saddle, then did all the rest themself?
What if they’ve brought the horse up from a youngster and nobody else has shown him, but a professional rides him once or twice every week to keep him “tuned up”? What if the professional doesn’t routinely ride him, but gets on once in a while when the owner is struggling? What if no other trainers have ever been on the horse, but the owner lets other amateurs or kids ride him some, maybe even show him?