I think you miss the point in your attempt to understand the use of 上 to represent on, above or over.
The Chinese language proves there is no need for more than one word to do this as the necessary information is provided entirely in the context of the conversation.
For example, I am thinking of sentences that may appear difficult:
"The bird was on his head", "The bird was over his head"
Which could both be translated in Chinese to something like:
Out of context you may try and guess whether it is right on his head or right above his head, however it could be either.
Here are some examples of how this could be expanded (taken from Google results):
有一只鸟在他头上飞过 — extended, no contact, vertical
A bird flew over his head.
鸟在他头上做窝 — contact
A bird built a nest on his head.
鸟在他头上飞了几圈 — extended, no contact, vertical
The bird circled above his head a few times.
I would be interested to find an example sentence in English that can not be easily understood in context in the Chinese with the same amount of information.