One example certainly isn't enough, but this one example here might be very helpful.
In the sentence above, there are two instances of the spelling "one". The first acts as a determiner. The second doesn't, since that role is filled by the word "this".
One way to look at this is to say that there are several different words "one". From this perspective, one could imagine a set of homonyms, each differing from the others perhaps only in word class: including the determiner "one", the plain adjective "one", the pronoun "one", and the common noun "one".
That perspective exists, but personally I don't like it. It lacks parsimony. It adds a complication that seems to lead more often to confusion than clarity.
I prefer to describe "one" as a pair of homonyms, an adjective and a noun. From my perspective, the adjective "one" can act as a determiner or as a substantive, meaning I need only one part of speech to explain the plain-adjective-like, determiner-like and pronoun-like behaviors of this one word.
My book is that one.
Here, I describe "my" as a personal pronoun, specifically the singular first-person attributive genitive pronoun.
Oddly enough, personal pronoun doesn't appear among the options you've considered.
Even the page you cite notes that "some grammarians consider determiners to be a part of adjectives." Merriam-Webster grammarians are apparently among them.
There is no real conflict between Webster calling it an adjective and Collins calling it a determiner.
From my perspective, determiner is a semantic role, not a grammatical word class. It's a role predominantly filled by adjectives, and sometimes filled by genitives. From my perspective, there is no conflict between this "my" being both a determiner and a pronoun.
How someone with a different perspective might otherwise resolve the apparent conflict is not something that I can explain.
