I agree, it will act the same if it isn't engaged all the way. However, I would eliminate switch problem because if the switch goes up and not down then you wouldn't be able to push back headrest since the lock would stay in highest position. Unless you pound on it and strip the gear and rail, but then it wouldn't go up again.
BTW engaging shouldn't take too much force, because it is angled. The headrest is designed to engage easily without putting in highest position and pressing button. My point is, it should have engaged when you were pressing down on headrest. Try putting headrest up a little and then press on it or put in highest position and then press button and try to engage. If it doesn't the lock mechanism is defective.