grammar

Is is possible to say that "Above" means "over" and "Under" means"below", but not the opposite?

According to my grammar:

Over - In a higher position (close or touching) Above - In a higher position

Under - In a lower position (close or touching) Below - In a lower position

Example:

There's a helicopter flying round above / over the school.

I would interpret this sentence as "above" meaning almost touching the school or not. And If I hear "over", the helicopter is almost touching school.

This makes me think about "over" and "on". "On" means that something is touching something, but if "over" can mean also "touch".Well, I would never use "over" to mean "touching something", I would use "on" instead.