"Over vs Above" and "Under vs below"

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.

There's a nice explanation in Webster's Learner's Dictionary. In many, many cases they're equivalent (although "over" is more common.) When they differ, it's because "above" is typically used only for location, while "over" can be used for movement from one place to anoher. Contrast "I climbed over the fence" with "I climbed above the fence." The first suggests I actually did climb the fence and now I'm somewhere else. With the second, it seems more likely I'm climbing something else and simply passed the level of the fence.

Under and below follow the same pattern.