why would the horse having eaten poison have any effect on it's execution? you said the rule is that EVERY SINGLE HORSE no matter what, no matter the reason if it throws it's rider just once it get's put down, you are the emperors son yet not even you can stop it so why would this change it?
honestly this rule is probably the stupidest thing so far, horses are extremely expensive to raise, they take time, effort and a lot of money to raise into war horses. horses throw of their riders, it happens, even expert riders with world class horses get thrown of now and then, it could be from the horse getting spooked, a misplaced saddle, rider error or something els. to spend all that money, time and effort to raise the horse only to execute it if it makes a single mistake is just stupid beyond belief.