OK, the racing saddle and the jumping saddle only become available after the horse has learned the skill. Just clicked on Prince and chose saddle, now there is the option for everyday, leading (no saddle) racing and jumping. The jumping option was not there yesterday when I saddled him up to practice jumping for the first time.
As for it making a difference if you change saddles, it probably does, but I've still won races using an everyday saddle.