"Sometimes people really do not mean to sound rude, they are just freaking out."
Yes, this, or if not freaking out, really having to concentrate. Generally I don't mean to sound abrupt but the horse I'm riding might just have decided there is a horse-eating monster in the hedge and be focused on that, meaning I have my hands full of a 600kg animal that at any moment might suddenly refocus, see your dog, and then decide to do something about the approach of a random predator. In which case "can you get your dog on a lead" in a rather abrupt tone is simply me trying to keep you, your dog, me and my horse safe. And a rider can feel if a horse is tense and about to arse around when to someone non-horsey on the ground it looks perfectly normal.
As for the instructions to dismount and lead your horse - just no. Sitting on my horse I can ask for leg yield to get his attention and regain control. I can say, with every fibre of my body "it's OK, I'm with you, the world is safe". If I dismount I've just said to him "help I'm so scared" thereby spooking him even more and giving myself a lot less control.
My horse is very, very good with dogs. He's hunted and he lives with dogs. He's had a mastiff gallop under his belly whilst the farrier had his foot up on a tripod and he did not care - that's about as good as it gets. He's a flight animal and even when his flight is restricted by his leg being held, he's still OK with dogs. But I still think it would be safer for your dog to be kept away from my horse on a bridle path, the one place I can chill out and relax with him whilst hacking.