I guess you're just supposed to learn and move on. And to be honest, you might remember it better because it frustrated you.
It's not the first thing you should come up with, but it IS correct.
If someone said to you: vorrei bevere l'acqua, and you said, io no or no io, it would be the same as "I don't" in English.
I understand that "io no" can be translated as "not I" as well as "not me," but I was wondering if there was a way to differentiate between the two; i.e., how might I say "not me" instead of "not I," or is there really no difference in Italian?
The difference between "not me" and "not I" is the case; the former is in the accusative while the latter is in the nominative. In Italian, as far as I understand it, there is no case, so there is no difference between the two.
There is a difference, though! "Io" is as different from "mi" as "I" is different from "me."
No. No in Italian is the same as "no" or "not" in English. The problem here is that IO is not "me". It's "I".
In another comment where "non" is used, it was said that "no" is "no" where "non" is "not". Is this not true?