2012 Trance X4 is 9 speed. So if you’re going to keep the rest of the drivetrain and just upgrade the RD and shifters, then you need to stick to 9 speed components.
RD-M770 is 9 speed, conventional profile, and rapid rise action.
RD-M771 is 9 speed, conventional profile and normal action
RD-M772 is 9 speed, shadow profile and normal action
Conventional profile RDs stick out a bit more than the shadow, so can be more prone to hits.
Rapid rise action means the spring tension returns the RD to the largest sprocket, rather than the more normal returning it to the smallest.
All come in SGS long cage or GS medium cage versions. For your triple cranks you need teh SGS long cage.
None of the 9 speed mechs have the new shadow+ clutch, that’s only available on ten speed.
If you don’t know the difference or don’t care, then safest bet is to just get the M772 SGS.
I’ve not heard of M772 shifters, and I’m guessing you mean the M770. These will match any of the above RDs, but beware there is an M770-10 (ten speed) so make sure you get the 9 speed version. Stainless steel inner cables will come with the shifters, and your existing outers may be ok, or if not buy a few metres of SP-41 outer to replace it.
If you’re going to the expense of replacing half the drivetrain, and maybe your cassette and chainrings are getting a bit worn too, then why not consider going the whole way and get a 10 speed XT groupset from one of the online stores in Germany for about 330 quid?