A good question, to which there is, however, no very simple answer!
Firstly, regarding the classification adjective vs. pronoun, among grammarians supporters of both are to be found. The argument favouring the classification 'pronoun' (probably the more traditional of the two) is the simple - and essentially irrefutable - one that words such as 'my' etc. share privilege of occurrence with possessive-case nouns (e.g. 'his dog' could replace 'Peter's dog'). Any form that can replace a noun should logically be categorizable as a pronoun.
The "rival" argument for the classification 'adjective' is twofold - one historical, the other practical. The historical argument is that the word-group in question once took adjectival terminations, making them agree with their noun referent in case, number and gender (as they still do in cognate languages such as German [meine, meinen, etc.]). The fact that such terminations happen to have been lost over time does not - by the lights of proponents - constitute a basis for reclassification. The practical argument is that, by terming 'my' etc. adjectives, we may thereby reserve the term 'pronoun' to distinguish the syntactically different group 'mine' etc. (which are purely, and uncontroversially, pronominal).
As for the classification 'determiner' - probably the most common of all in contemporary works of reference - this does not really directly conflict either with the pronoun or adjective classifications, since determiners, rather than a simple form-class, constitute a higher-level function group (a sort of 'super-category', one might say) which cuts across a number of form-classes, ranging from possessive-case nouns to articles.
I hope that the above makes the issue a little clearer for you (and enables you to see why, in various different ways/senses, all three classifications can be justified!)