Any form of shoulder press will work all 3 heads of the deltoid, but mainly the anterior and lateral. Varying angles, positions, grips can shift the focus more onto a specific head.
A very basic way to prove this is to put your fingers on the anterior delt while you mimic the movement. Can you feel the muscle contracting? Yes, so it is being worked.
As regards your range of motion, I don't bring the bar any lower than eye level, for the same reasons pumpster already mentioned.
Tell this 'trainer' to politely piss off, if he keeps interfering, make a complaint.