Terminological illogicality: possessive pronouns and adjectives

Enough of yesterday's self-indulgent self-pity, and back to the business of righting all the wrongs in language teaching.

Last night I was standing outside my tent and for some reason or another I started thinking about possessive pronouns and adjectives.  Now when I first encountered this terminology, the way I was being taught them seemed wrong.  Well, I've thought about it, and they were wrong, because the terminology was being applied in terms of other languages, not English.

If you're not familiar with the terms, I'll just define them by example: my is a possessive "adjective" and mine is a possessive pronoun, by standard nomenclature.

But now to demolish standard nomenclature.

Possessive adjectives
"My" is considered an adjective because it goes before a noun and gives additional information.  "My bike", for example.  But... in English, countable nouns never occur in the singular without some kind of determiner, whether that's a number ("one car"), an article ("the car", "a car") or a demonstrative ("this car", "that car").  Shouldn't we define "my" as a possessive determiner then?

Perhaps, and that used to be my preferred solution.  But the more I heard native speakers say that they were convinced it was a pronoun, the more I was convinced it was a pronoun in the native model.

Consider, in "my bike", the word my substitutes for Niall's - a noun in the possessive case.  Something that substitutes for a possessive noun should be called a possessive pronoun, surely?  So as I said already, I agree that "my" etc are indeed pronouns within the grammar of English itself.

The reason this isn't the case in traditional grammar can only be described in terms of other languages.  French and Spanish don't have a "possessive case" and instead have to say "the bike of Niall" (le vélo de Níall/la bici de Níall).  I would say then that French and Spanish use their possessives as determiners, because "the bike of Níall" goes to "his bike", just as "the bike over there" goes to "that bike".  Determiner.

But this isn't the case in Italian or Catalan, where the article is retained in the possessive "il suo bici" -- "the his bike".  So in Italian and Catalan, perhaps it is an adjective, but that doesn't mean we should try to call it one in English, French or Spanish... or any other language for that matter.

Possessive pronouns
Ok, so what about "mine", which is traditionally called a possessive pronoun?

It gets that name because it allegedly substitutes for a full noun phrase -- "that is my bike" becomes "that is mine".  However, that's all well and good where you have a language where adjectives can be used pronominally, but that's not the case in English where an adjective is often accompanied by the pronoun "one".  No, not one as in "one must", but one as in "a green one".

In English, we use that same pronominal "one" with possessives to prevent redundant repetition of the noun -- we can and do talk about "my one", "your one" etc.  Yes, anywhere where we can use "my one", we can substitute "mine", but doesn't that shift the meaning significantly?

When I say "mine", am I talking about the thing anymore?  Or am I instead talking about a property of the thing? -- its mine-ness, or its belonging-to-me-ness?  Because if it's a property -- and my gut reaction is that that's exactly what it is -- then it's an adjective.

Edit: [2015-02-19] My gut reaction was wrong. I hadn't thought about the use of "mine" etc as an object to a non-predicative verb -- eg "I'll show you mine if you show me yours."