Question: I notice you ride English in a Western saddle (photo below of you and Max). How does the horse know the difference?
Answer: Horses have no idea that one saddle is different from the other, or that one is for English and one is for Western, besides the feel and contact differences. Otherwise, I am using my aids exactly the same for both disciplines.
Only humans think of the differences. Horses can only ‘feel’ them.
Before I get discipline specific with a horse, I train them to the point where they are soft, responsive, have cadence and rhythm, understand aids and have really great stop and go buttons. Then I will move them into the disciplines and refine the horse from there. I would not train a hunter the way I would train a reining horse, once the basic structure is in place.
Max in balance
I also find value in having my horses comfortable with various forms of saddles on their backs.
I often wear breeches and boots to ride Western as they are comfortable, and when I ride both disciplines on the same day, it’s easier to ride Western in English attire than English in Western attire.
The main thing is the training. Max goes English and Western, so to him, as long as the saddle is comfortable, he’s fine. I have often ridden reining patterns at home in an English saddle. I also ride bareback once a week, to maintain balance and it’s valuable for finding any bad habits you may have picked up. Bareback riding demands that you are centered, balanced and flexible.
Read about the advantages of bareback riding, on Horseman’s U.com
aprilreeveshorsetraining 
That’s a good idea to make sure that your horse could be comfortable in both kinds of saddles. I would think that would be useful if they had to use the other kind for whatever reason for a while. I’ll have to consider getting one of each so that I could get my horse comfortable with them if I decide to get one.
And a great idea for a rider! I get on student’s horses all the time, and never alter the stirrups, so you learn fast how to sit in anything. But the horse too needs as big an education as you can give them. Learning about horses is endless, which is one reason I gravitate to them so deeply.