In your case, it doesn't make much difference whether you choose plywood or OSB. Plywood is more expensive but nicer to work with. Screws hold slightly better in plywood than in OSB, but the difference is minor.
12 mm or thicker? As a reference, it can be mentioned that 15 mm plywood is approved behind gypsum in bathrooms to allow for screwing things up anywhere. This means you don't have to hit the studs when hanging cabinets, shower walls, etc. If you choose 15 mm that is well screwed into the studs, you can screw the rail anywhere, even though it's best to hit the studs, of course.