Nope, my all-arounders always knew the difference.
The problem is, you don't want to "shut down" the western jog or the western lope. You actually want the horse to keep suspension, but at a slower pace or shorter stride.
Think of your horse as a spring. When you slow jog, you want your horse to still feel as if he is moving forward, keeping the suspension, but his is actively collecting and pacing himself slower. The spring is tighter and shorter, but the energy is still in the spring, and the spring can still be pushed open if need be.
Then when you push your horse into the hunter trot, you are releasing the spring and asking him to lengthen his body and his stride - while still keeping his collection from before.
Typically, I'll do exercises when I'll school both western and hunter, because they help each other out. Doing the hunter trot keeps the western jog moving forward and suspended, and doing the western jog helps keep the horse collected up and using his core in the hunter. I will do figure 8s using the entire arena. I will encourage my horse to jog through the short ends, and then I will open up my horse to hunter trot through the long middle portion. My goal is to control my horse's stride without losing forward motion, suspension, or collection. And I don't change tack. My horse knows even in a western saddle to hunter trot or western jog. Being able to transition between those two speeds is extremely important in training, especially if you find yourself in a horsemanship or equitation class where they ask you to transition from sitting trot/jog to forward trot. Or if you are in a trail class and the layout of the poles asks for your horse to extend his natural stride without being obvious to the judges.