Seamus is exactly right about red coats in show jumping.
Red coats in fox hunting are different, and have nothing to do with being blooded. Red coats are worn by paid members of the staff, such as the huntsman and whippers in, and by male members of the hunt who have earned their colors. Earning your colors allows you to wear the red coat with the hunt's colors on the collar and the hunt's buttons. Traditionally, women did not wear the red coat, but just the collar and buttons, unless they were also hunt staff, but that is changing. You used to be able to tell the difference between professional hunt staff and members by the way they wore the ribbons on their hunt caps, but safety helmets have done away with that tradition too. Hard riding junior members can be awarded their buttons as sort of a half step towards earning their colors, a junior who earns their buttons usually earns full colors shortly after their 18th birthday.
Earning colors is supposed to indicate competence in the hunt field and service and dedication to the hunt club, but there's always some nasty gossip and speculation about whose "service" was writing a big check. It usually takes several years of regular hunting and work with the club to earn colors, sometimes more. But the core idea that a red coat means someone of experience and authority is sound, you just shouldn't overlook the lady with just the collar and buttons, or the junior with just buttons, they may have just as much experience.