The two are very different phases, so yes, they are ridden differently - same way as you would not ride “dressage-like” in an XC field. That being said, there is also factoring in optimum time for both phases.
I don’t usually ride stadium in 2-point. Since a stadium round is all about being clear and correct and in the time, I ride the canter to get the jump/distance I want. The canter I want for my horse is very different than the canter I want XC. I want a “Showjump Canter” - I want a bit more ‘collection’ (I use that word loosely), a more compact stride and lots of amounts of adjustability. A crappy canter will lead to a crappy takeoff and crappy jump and crappy landing, so there is much more monitoring of the canter going on for me in SJ. I am not sitting as deep in the saddle during SJ as I would be in DR, but I am not getting off the horse’s back the way I would in XC. Since the efforts are usually fewer and the time shorter in SJ I work for a powerful, showjumpy canter - which for most of my TBs, requires more effort than XC-type canter. In stadium rounds the jumps come up quick so it is very important to establish the right rhythm right off the bat. In SJ my speed is very consistent - there isn’t any going from 250 to 375. Depending on the level, I’ll pick the rhythm I need and maintain it. Since most stadium rounds are on footing or reasonably flat surfaces, my position is mostly upright without two-point or leaning too far back.
XC is a completely different animal where I tend to let the horse canter the way they would naturally since there as much, much more time to set up in between fences. So the mpm changes - some spots we might be going 275, and others we might need to really step on the gas. I am all about conserving the energy, especially on XC course, so I don’t like to do a lot of picking at their strides unless we are about to get buried deep in front of a fence. I want a fluid, efficient canter + gallop. I want to keep them off of their forehand, but I am not trying to get them to have that super uphill, super collected show-jump canter out on terrain. Speaking of terrain, since the terrain is changing my position is changing with it - the entire XC run I am focused on keeping myself up off their back - right before fences I will start to sit deep in the saddle and that is a cue in itself for my horses to slow down and prepare for the fence.
There are also different types of jumps that require different types of positions over them; IE you ride a bank down differently than you’d ride, say a table. For most XC fences you really want to be in a driving type of seat - you don’t want to be ahead of the motion, which for a HP can be a hard thing to adjust to. Think about keeping your shoulders and pelvis level, and rather than fold over the horse’s neck over the fence, think about having his withers brought to your chest.
You’ll get it! I am also an ex-HP transplant, though I made the discipline switch nearly two decades ago. Part of it is muscle memory, part of it is like anything else: practice makes perfect and exposure begets experience 