I agree with Barry. The horse will adjust to what it is trained for, but traditional western riding uses different tack and different cues. My Appy gelding is an ex-ranch horse, and he would NOT be happy if you wanted to ride him with contact. If you wanted to put my Arabian mare 'on the bit', you had better be ready to train her for it or you will have one pissed off mare.
It is entirely possible to train a horse to do both, and do both well. Once trained, you are free to mix and match at will. But traditional western riding revolved around ranch work, and that generally means one hand, slack reins and much less emphasis on collection (and pretty much never a collected gait).
Traditional western riding: Emory Sager, of the Shoe Bar Ranch, on "Old Blue", his "wonder horse", cutting an animal from the herd. Shoe Bar Ranch, Texas
, 1912 Erwin E. Smith Collection Guide | Collection Guide
Of course, that is not a whole lot like modern Western Pleasure either. WP is much closer to dressage than what I've seen on ranches or from trail riders - which is pretty close to the B&W photos from the early 1900s!
As for cues, I remember the first time I ask some western riders what cue they used to get their horse to canter. They all looked at me like I was from Mars, and then one said, "Um...kick harder?" Horses respond to how they are trained. If someone riding Mia makes a kissing sound, they'd better be ready to shift up a gait. That isn't really standard anything, but it is how she has been trained.
English or Western, I think a rider ought to ask about the horse rather than assume anything. You can probably assume a lot about a properly trained dressage or cutting horse, but it would be a bad idea to assume the cutting horse will respond like the dressage horse. And if the horse is privately owned and ridden, like Mia, then ALL bets are off...