Rob_E

Yes, this is incorrect. Or, rather, this is not necessarily correct. Vegan is a dietary restriction in which you don't eat animals or animal by-products (like eggs and dairy products). Many vegans adopt this diet for reasons involving animal exploitation, so they likely would not be a fan of horse racing and similar uses of animals for non-food purposes. Many vegans adopt this diet for health reasons. They might be fine with horse racing, kicking puppies, or stomping on baby sea turtles as the crawl from their eggs to the ocean. Probably not all those things, but the fact that they don't eat animal products really doesn't carry any information about their morality or beliefs.

Likewise vegetarians don't eat meat, but might eat animal by-products. They might have multiple reasons, some involving animal exploitation, some not.