It doesn't seem ambiguous to me, it seems like "của mình" is used when the object belongs to the same subject of the sentence. She wants to drink her own water in this sentence, "của mình" kind of takes "her own"'s place.
I don't know Vietnamese, but if my understanding of this concept is correct, "Cô ấy muốn uống nước của cô ấy" could mean she wanted to drink another (female) person's water. (There's a similar concept in the Scandinavian languages.)
In this given sentence, the object (water) would belong only to the subject (she).
If the speaker wanted to say "She wants to drink my water", it would not read "Cô ấy muốn uống nước của mình." I don't know how exactly it would read, but it would not be that, because "của mình" doesn't exclusively mean "my" or "mine"... Here's part of what the notes say:
"To avoid [repetition], one uses của mình. Here is the simple rule: của mình can replace any possessive adjectives or possessive pronouns above if and only if the noun(s) (in the sentence) belong to the same subject (of that sentence)."
@Camytrang. actually no, I would not use "cô ấy" and "nó" in reference to the same person. "cô ấy" is a formal and polite way to refer to a young woman, usually someone that has no connection with you. for instance, you won't call your sister "cô ấy". on the other hand, "nó" is used to refer to someone younger, of lower status, or whom you look down on. your sentence sounds to imply two different people, the woman who's drinking the water and some other unnamed person whose the water belongs to.
As a native speaker I'd have to agree that "của mình" definitely means "(of) mine." The possessive statement of "her water" ought to be "nước của cô ấy," though I find it to be unnecessary to refer to whom the water belongs as it is too long for casual conversation and a peripheral detail at best.