not mind = be not above?

I came across the idiom "be not above (doing) something". But its meaning is either too long to understand or a little logically complicated.

From Cam:

to be willing to do something, especially something that is not what people expect you to do or think you should do:

From Longman:

not be too good or honest to do something

From Longman:

not mind doing something: to be willing to do something

Can I draw the conclusion "not mind" = "be not above" from the perspective of meaning for simplicity?

Eg: He's not above lying to protect himself = He doesn't mind lying to protect himself.

But it seems that what follows "mind" mostly refers to something that is done by someone but is neutral or even beneficial for someone else. What comes after "be above" is usually something good for someone but bad for someone else.

Eg:

Would you mind closing the window? It's too noisy outside.

No, I don't mind (closing) it.

How to get a fair conclusion about this comparison?

Consider a king. Does the king have to sweep the streets in his country? No he is too senior for such a dirty job. You might say that the task is "below his status" or that he is "above" that task.

On the other hand a person might say that they are willing to do a dirty job. Perhaps because we value people who do hard work, or because they believe they would do the dirty job well.

There are lots of ways a job might be unpleasant and "dirty", and though suitable for a "low status" person. It might be physically dirty, or it might involve deception or criminality.

So if I say "He's not above X", where X is some unpleasant task (either physically or morally unpleasant) I mean that he is willing to do X despite the perception of him as a high-status person.

High-status, here, means noble, or rich, or honest.

So it means a little more than "don't mind". In your example, "closing the window" is not a particularly dirty or unpleasant task, so you would not say "I'm not above closing the window".

On the other hand, "lying" is unpleasant, and not something you would expect a high-status person to do.

I'm not above some creative accountancy, if it means the company doesn't go bust.

"Creative accountancy" means submitting deceptive accounts. An honest person would not do it.

I might be a famous garden designer, but I'm not above doing the weeding.

Weeding is a low-status garden task, designer is a high-status position.