Are pronominal adjectives and determiners just different terms to describe the same thing? If not, how do they differ?

I    (source of the paraphrasing below: CGEL)

Fundamental in the concept of determiner, the article is a particular type of determiner called central determiner (three forms: a, an, the). There are two other types of determiners: the predeterminers and the postdeterminers.

• She followed her father in his travels to all the many faraway countries where he had to work. (1)

Pronominal adjectives are words such as "his", "this" and "some" which are called pronominal adjectives because they are both adjectives and pronouns depending on the word environment. They could have been called as well adjectival pronouns. This is just a term that describes their double function on the basis of a single meaning; however only one of the two functions can be effective in any given word context.

• His (det.) hair is dark; the girl's hair is dark, his (pron.) is not. (2)
• The cat drank some (det.) milk. Did you pour some (pron.) for the dog too? (3)

The pronominal adjectives are determiners or more precisely central determiners, this being so because they occupy the same position in the noun phrase that the articles do, the articles being the most basic central determiners.
This means that the article is not the only possibility for determining nouns in a fundamental manner. Instead of "a" or "the" you may find "this", "that", "every", "each" and so on. These words form a set of closed-class items (finite in number) that are mutually exclusive with one another (for instance "a the boy" and "some the boy" are not possible combinations). They constitute another sort of central determiner that differs from the basic ones in the following ways.

• Whereas the articles have no function other than making precise how the head is used, the pronominal adjectives have the function of head in a noun phrase. This is shown in the sentences "(2)" and "(3)". The central determiners "no" and "every" are exceptions; they cannot function as pronouns.
• A second important difference of the pronominal adjectives is that unlike the basic central determiners, they have a lexical meaning.
• An important number of them can be used with "of".

II

In the 2nd table found on this page (Cambridge Grammar), the 3rd and 4th column, labelled respectively "demonstratives" and "possessives", show two classes of pronominal adjectives; the second table shows the class of pronominal adjectives that can be used with "of". Those tables show all the central determiners that are not articles or in other words all the pronominal adjectives. Note that the terminology "pronominal adjective" is not used in CGEL nor in the Cambridge Grammar and what is called "adjective" in the second column of the first table has nothing to do with pronominal adjectives; those latter are not found in that column.

III

"Other" is a postdeterminer and "every" a central determiner (as you must have noticed in the above). "Another" is a central determiner (you can combine it with "of", it's not so for "other").

In "She had eggs almost every day." "almost" is not a determiner but an adverb although some grammarians consider words of this type to be determiners (ref.).

3 April 2021    Reply to comments that I apparently failed to consider (user BillJ, user tangosquared)

I just became aware of your unanswered comments. The CoGEL treats predeterminers as determiners (Chapter 5    Nouns and determiners). Modification (at least in CoGEL) is a concept that pertains to the noun phrase among other things: it connotes particular intrinsic characteristics of the referent of the head. It is achieved for instance through copular relations and the use of attributive adjectives. In "all the white cards" the adjective "white" modifies, or even more precisely, specifies a sub-sort within a sort; on the other hand all does not contribute to making precise the nature of the cards itself, it does not modify the general idea of "card" nor the general idea of "white card" for that matter; rather, within the given sort (white ones), it allows to determine more certainly which white cards one has in mind ("none is being left out" is the determination). This is the idea in the difference between "determination" and "modification" and in my opinion the definition (implicit in CoGEL) and the corresponding choice of term (in CoGEL) are fairly sound. So, we are confronted to a question of terminology, of naming, and there is no great problem as long it can be asserted that "modification", as used in CGEL and CoGEL corresponds to different definitions (I doubt that though).
As the idea in what perform words of the type of "all" in the noun phrase is similar to what do words more evidently of a determiner type such as "the" and "a" they are reckoned with as determiners; as, among other characteristics, they show the syntactic one of appearing sequentially before the central determiners (a, the,…), they are called predeterminers.