A couple of years ago, I wrote a post describing my objections to the traditional definitions of possessive adjectives and possessive pronouns. At the time, I still favoured calling what is called the possessive pronoun a "possessive adjective" (eg. "it is mine"), and calling the possessive adjective a "possessive pronoun" (eg "my car"0 for English... in theory.
However, practicality is another thing entirely, because there is nothing more confusing than using the same words as someone else to mean entirely the opposite thing, so I have never used the terms that way for students. In fact, I actively avoid using either term if I can possibly avoid it, as it doesn't seem helpful.
So recently I've been working on trying to find ways to better categorise grammar, and I've settled on what actually seems like a reasonable compromise.
For the possessive "adjective" of traditional grammar, I'm going with the alternative from the previous post -- the possessive determiner. It aligns with this, that, a and the, so it naturally falls into the class of determiners. This doesn't mean it isn't a pronominal form -- it actually means that forms like "John's" have to be considered determiners themselves... which is entirely logical, as "John's car" is "the car that belongs to John", but "John's" has replaced the definite article; hence "John's" must be a determiner anyway.
For the possessive "pronoun", I'm going with the possessive predicate. I decided on this when I was thinking about it as a predicative adjective in sentences like "It is mine," or "that's yours." Of course, that's not the only situation it occurs in, which is something I overlooked a little when blogging from the unbearable heat of Corsica in the summer. There is no predicate in "I'll give you mine," or "She didn't have hers, so she took yours." But at least it gets away from the counter-intuitive implication that the other form is not a pronoun.
I will continue to think about this, but if I do ever come up with a better term, I can just do a search-and-replace on what I've already got without any problems....