Well, the problem is that as you practice, your subconscious mechanism picks up repeated tasks. If you are still "fully present" (I'd call it "trying hard") once your subconscious develops capability, you end up with two brain systems struggling for control of the muscles of the body.
You can see this clearly when you start learning a new technique. There's usually a period when the technique starts improving quickly-- as you get a conscious grasp of it over a short time. But then it gets worse again, and you wonder how you lost the magic. My theory is that this is because of the conflict I mentioned.
The next stage, where I think most people either fail or take too long, is to let go of conscious control on the physics of playing-- you have to have faith in your body to do it. You should be "fully present" for performance variables-- new rooms, new sounds, the mood of an audience, getting to know new bandmates, etc. You shouldn't be projecting your ego and soul into just your hands.
You aren't "fully present" when you walk. Try it-- "Left leg-- push it forward. . . right leg. . .move it now. . . oops forgot about the left leg." It would be a disaster!
That's what your teachers are trying to do-- engage your conscious mind on an external stimulus so that you can't be "fully present," which in the end is another way of saying "not trusting the subconscious, so keeping a conscious death grip on the body."
You can use ANY kind of distraction-- watching a movie, meditating on the breath coming in and out of your nose, reading a magazine. You can even try falling asleep. If you can fall half-asleep but be JUST conscious enough to keep playing, you turn into a music god. It's hard to achieve though.