As far as the chart you reproduce is concerned I think it's misleading, there's no difference in position between beneath and under and both should be on the same level.
I would say that beneath and under are near synonyms, they both mean that something is lower than something else. The essential difference is in register.
Beneath is more formal, more literary.
The choice of one over the other depends on what you want to convey.
I'm wearing a sweater under my jacket.
I'm wearing a sweater beneath my jacket.
both mean that the jacket is on top of the sweater.
Some people might see a difference between:
(1) The ground under my feet.
(2) The ground beneath my feet.
1) referring to the actual piece of ground my feet are touching
2) referring to the ground in general, or a larger piece of ground than just the actual piece my feet are in contact with.
I would say it is context dependent and I would use beneath in both cases, and maybe not under if I meant the ground in general.
When used in an abstract way beneath is preferred to under.
- Myrtle Wilson (in the Great Gatsby), married beneath her class.
- If I find that something is beneath my dignity, it means I have such a high opinion of myself hat I could not do it without degrading myself.
But:
We have a lot of work at the moment and we are under great stress.
Note:
As in you examples I have only considered the use of both words as prepositions. But note that under is very rarely used as an adverb (I can't think of an example where it would be used as one - which does not mean it is impossible, of course) whereas underneath can.
The garden is just under/beneath my bedroom window. (preposition - it is followed by a noun)
When I look out of my window I can see the garden beneath/underneath. (adverb)
The sentence is ungrammatical because of the presence of "ever". "Ever" belongs to a group of words called "negative polarity items".
The unusual thing about negative polarity items is that they can only appear when there is some suggestion of doubt or unreality in the sentence.
Here are a few of the situations where you can use "ever":
- In an explicitly negative sentence. Note that "ever" becomes "never" when negated.
I have never done that.
- In an implicitly negative sentence, such as one expressing doubt or infrequency.
I hardly ever do that.
I doubt that I've ever done that.
- In a question (because the answer could be "no"!).
Have you ever done that?
- In a hypothetical situation.
If you ever do this, let me know.
There are a couple of other situations. See this answer on English Language & Usage for more detail.
Another negative polarity item that often causes difficulty is "any".
Best Answer
In that case, beneath is not the correct word. We would not say that the gum is sticking beneath the table.
We would say as you have said, that the gum is sticking to the underside of the table, or on the underside of the table. I know of no one word that means "on the underside" or "to the underside".
A teacher might say to the class:
Don't be sticking your chewing gum up under your desks!