Let's flip this around: once you learn to drive, and then drive enough for it to become something you do unconsciously, is the car now part of "you" too?
If not, then I wouldn't use the embodiment/integration argument to define where "you" is, as the brain can learn to turn just about anything into extra limbs or senses, if you use it frequently enough.
If not, then I wouldn't use the embodiment/integration argument to define where "you" is, as the brain can learn to turn just about anything into extra limbs or senses, if you use it frequently enough.