Is either "can" or "could" more polite?

Version1:

The problem is that "can you x" is like asking "are you able to x" whereas "could you x" is like asking "are you willing to x"

So yeah, politeness isn't the issue at all.

This is probably why, even if you're not aware of the nuanced meaning, could is more polite. It's not insulting the target by questioning their ability to perform something.

Version2:

To answer the question: "could" definitely sounds slightly more polite than "can" to a native speaker since it is less direct and more deferential as a result.

"Could" is a form of "can", so both are technically asking "are you able to...". This is not the difference between the two.

The difference is that "could" is used with the conditional mood in English. There is an unstated by implied "if" clause here. The sense of the sentence is, "if you weren't too busy at the moment, could you X?" This is why it is less direct and more deferential.

"Could" does not have the sense of "are you willing to X?". That is "would". I could also ask "would you X?" This is polite, don't worry about it in practical terms. If I had to think very hard about it, I'd say "would" is slightly more direct and assertive. It is questioning whether the person wants to do X. "Could" asks whether they can do X. And that is maybe more indirect. It turns the question away from "do you want to do this for me" to "are you in a position to do this for me at all?"