Yes, this is true. When I was taking veterinary classes years ago one of my instructors described horses as occupying a 'grey area', precisely because they are neither completely livestock animals nor are they truly "pets" (i.e. small animal pets like a cat or dog). It's because horses are still used as commodities that any Federal protections would come into play, so those racing, rodeo, etc. are kind of outliers these days, from the legal perspective. Peoples' pets also are not afforded coverage by AWA, but most states have at least some animal cruelty laws on the books, which can give them some coverage in cases that warrant protections.
Incidentally, about a decade ago I contacted a Federal official and inquired why horses used for racing and rodeos are not covered by AWA and never got a response. Aside from a lengthy amendment, I assume staffing is a real problem. They would also likely close down every race track int he country if they started doing inspections... I've seen the medical records for the Kentucky Derby horses in the past (the pinnacle of the sport) and most of them are abhorrent.