The language question: No. Every horse is different, and every horse is trained different. Unless you're always riding horses by the same trainer, almost every horse you ride will be different.
This is very true training-wise.
However, most trained horses are going to be motivated by the similar things body language-wise. For instance, most horses, if you swing something at their hindquarters and cluck, will shift away from you. Most horses, if they haven't already decided you have no leadership over them, if you act big and say "ah-ah-ah" will kinda look at you to assess the level of threat you are presenting. Depending on the horse, they might even bolt away if that's what their life expierence has taught them to do. For most horses, a good correcting touch is a short one - if you push a horse away, it doesn't work, but if you jab your finger into said horse (don't reccomend this in the flank/belly area), usually they'll move away.
Undersaddle "language"-wise (I would call that undersaddle cues, personally), it depends on what they've been trained to understand.
I think a lot of it might be regional too. I know here, in Oregon, I can get on pretty much any horse, squeeze my legs+cluck once for a walk, Squeeze+cluck multiple times for trot, and squeeze with one leg+kiss for canter, keep kissing for gallop.
But, in youtube videos, I have discovered that people in other places sometimes kiss for trot, sometimes cluck for canter - VERY strange.
Most horses have a similar foundation and most will half respond to whatever is closest to "their" cue. However, you probably won't get a great response unless you are using the really correct, for that horse, cue.
I know with my mare, I bought her without any discussion with her owner on what her cues are so I've spent the last 4 years discovering things that get her to do behaviors I had no idea she did!