Is a determiner considered an adjective or a separate part?

This answer is subject to which grammar source you use, but if I understand it correctly, determiners are members of a particular class of words which come before a noun or at the beginning of the noun phrase.

Determiners are:

• the definite article: the
• possessives: my, your, his, her, its; our, their
• demonstratives: this, that, these, those
• interrogatives: which, what, whose
• general determiners: a; an; any; another; other; what
• quantifiers: a few, a little, much, many, a lot of, most, some, any, enough, etc.
• numbers: one, ten, thirty, etc.
• distributives: all, both, half, either, neither, each, every
• difference words: other, another
• defining words: which, whose

Pre-determiners. They go before determiners, such as articles: such and what, half, rather, quite

Adjectives are words that modify nouns and pronouns, primarily by describing a particular quality of the word they are modifying.

Determiners can function adjectivally, as can some nouns that are found chiefly in fixed phrases where they immediately precede the noun they modify (bus station).

In the phrase, "Which of my twelve pretty white silk hats"' pretty and white are adjectives; silk is an adjectival noun, and which, my, and twelve are determiners preceding the adjectives.