Horses, cactus and people.
Most people want to avoid anything that could go badly. Avoid anything that MAY upset the horse. Avoid anything that may get them in trouble. Avoid TRAINING their horse to stay out of trouble. Teaching horses about cactus isn't really that hard. I don't recommend doing it mounted, however, I have on many occasions. Unless you have the stupidest horse on the planet, they are better about the learning than you may think. Don't shove their nose on it, simply present it for them to inquire about. If you've done a good job teaching your horse how to investigate new adventures and objects this goes well in most cases. They also need to learn about as many types of cactus as possible. Horses don't have a cactus book and don't care that it's in the same family. They need to be introduced to all the cactus that you have available or can find.
When your horse is in a frenzy and/or bolting through the terrain is not the time to go "Look out for that cactus." Of course, if he's listening at all (oh, but he's not because he's bolting), he would still be saying "What's a cactus?"
Do your homework before you need it. Get your 12' rope and let them investigate while you're not too close. Or, if your horse is pretty good about knowing that the real estate that you're standing on is never the place for him to be, you can do this right next to horse. But, do do it.