I'm missing a term: What do you call "mine"?

Ok, I think it's past time to show my ignorance. I've been using the term pronoun to refer to all the following words:
> I, thou, he, she, it, we, you, they
> My, thy/thine, his, her, its, our, your, their
> Me, thee, him, her, it, us, you, them
…but also
> Mine, thine, his, hers, ours, yours, theirs
Am I supposed to use a different word for that last set? Or are all of them pronouns but there's a term to further distinguish them? Anybody know?
Let me get even more explicit: Are both "my" and "mine" possessive pronouns? Or is "mine" something else?
According to the article on "possessive adjective" at Wikipedia, both "my" and "mine" are possessive pronouns; they're further distinguished as "weak" or "dependent" ("my"), and "strong" or "independent" ("mine").
The "adjective" in the article title reflects the way both forms can be used grammatically:
This is red cloth.
This is my cloth.
and:
This hammer is large.
This hammer is mine.
(The article also suggests that possessive pronouns are "determiners" rather than true "adjectives" because, like such words as "this", they can't be combined with articles; you can't say "On the table is a my book" or "The his story is sad".)

All of those words are pronouns, but if you want to sound grammar-nerdy (and, really, who on this message board doesn't?) when you talk about them, Bob Bridges, there are three different names for these different pronoun groups:
"> I, thou, he, she, it, we, you, they" are nominative-case pronouns (or, probably more correct, pronouns in the nominative case);
"> My, thy/thine, his, her, its, our, your, their" are possessive pronouns;
"> Me, thee, him, her, it, us, you, them" are objective-case pronouns; and
"> Mine, thine, his, hers, ours, yours, theirs" are possessive pronouns.
However, I have seen the group my, your, his, her, its, our, and their called possessive adjectives, rather than possessive pronouns, likely because of the position of each directly before a noun. I've also found that distinction useful in the past for teaching English to speakers of other languages, since the Latin languages have similar usages. It's easier to keep separate the different usages of the words my and mine to Spanish-speakers by comparing each to the Spanish equivalent — for example, "este es mi cacahuete" ("this is my peanut") and "este cacahuete es mio" ("this peanut is mine"). (I'm not sure how to make accent marks on this board, so my examples are missing a couple — over the initial "e" in "este" and the "i" in "mio"; also, I used the Spanish word for "peanut" because "cacahuete" is one of the most fun words to say in any language.)

Yeah, "cacahuete" is a trip, isn't it, for some reason? It's a new-world word, of course; it never came from Spanish or Arabic, must be something from Aztec, Incan or some such.
The only thing that worries me—the reason I asked the question in the first place—is that "my", "mi", "mon" etc are adjectives and "mine", "mio", "mien" and so on are nouns. I figure there ought to be different terms for them. I mean, we have "relative pronouns". Maybe "mine" and "thine" are "substantive pronouns", or something, to set them apart from "my" and "thy"?
Funny how Spanish shares with English the little quirk of using the same word for both meanings of "this" in tunawrites's two peanut sentences. In Japanese, they'd be:
Kore wa watashi no tawara desu. ("This is my peanut")
Kono tawara wa watashi no desu. ("This peanut is mine")