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.

Wikipedia defines determiner / determinative (the POS, not the grammatical relation) in largely semantic terms:

A determiner, also called determinative (abbreviated DET), is a word, phrase, or affix that occurs together with a noun or noun phrase and serves to express the reference of that noun or noun phrase in the context.

A clear-cut example of what 'in the context' might imply is

"Pass me that book, please." [pointing to one]

Obviously, nothing about inherent properties of the book (weight, colour of binding, value, readability, genre ...) is being specified – merely its position in the environment in relation to the speaker's index finger. Its reference in the relevant context.

"That book is heavy", on the other hand, is giving an inherent attribute of the book by using 'heavy'. This is an adjectival usage.

But there are grey areas.

"Pass me the red / smaller / largest book."

use 'red' / 'smaller' / 'largest' as identifiers within the context of all the books in sight. ('Polar' in 'polar bears' is a classifier, 'red' in 'Her car is red' is a descriptor.) Though this overlaps with determiner usage, it's still reckoned that 'red', as it uses an inherent property of the object to identify it, is an adjective here.

"He is a mere youth."

uses 'mere' as a descriptor of the whole class to which 'he' belongs: immature, physically not yet at their prime, not ready for great responsibility.... It describes the context, but behaves adjectivally, giving attributes largely found in the whole group. It has been called a 'non-semantically-predicative adjective'.

In order to understand what 'determiner' as a grammatical category means, you need to understand the difference between a noun phrase (NP) and a nominal.

(1) My new car was stolen.

The subject of (1) is in the form of an NP wherein 'My' acts as a determiner and 'new car' is a nominal. So in a nutshell, when you add a determiner to a nominal, you get a full-fledged NP. I say "full-fledged" because often a nominal cannot be a full-fledged NP as follows:

(2) *New car was stolen.

Sentence (2) is ungrammatical because the nominal 'new car' is merely a nominal and cannot function as a full-fledged NP. Here, 'new' is definitely an adjective that modifies the noun 'car'. But having an adjective such as 'new' cannot make a nominal an full-fledged NP, because the nominal 'new car' simply denotes a rather abstract concept when in context the sentence needs a concrete entity as its subject.

Sometimes, though, a nominal can be a full-fledged NP as follows:

(3) New cars were stolen.

Here, the plural noun 'cars' in and of itself denotes concrete entities; therefore, the nominal 'new cars' itself can be a full-fledged NP.

In which case, of course, you wouldn't need any adjective either:

(3) Cars were stolen.