You’re definitely right that “mine” functions as a pronoun in “She is an old acquaintance of mine.” It means “that which belongs to me,” and has the same meaning as the adjective “my” when used with a noun that follows it, as in “She is my old acquaintance.”
In “This table is mine,” however, “mine” works not as a pronoun but as the possessive case of “I” functioning as a predicate adjective that modifies the subject of the sentence. In that particular sentence, the subject is “this table” and the predicate adjective is “mine,” which is connected to the subject by the linking verb “is.”
You also asked whether the word “mine” is a determiner in “The table is mine.” The answer is no. But “mine” has a special function as a determiner that’s now considered archaic. In poetry, it can substitute for “my” before a word that begins with a vowel or a silent “h,” as in these two lines from Sir Philip Sidney’s “Sonnets from Astrophel and Stella”: Let not mine eyes be hell-driven from that light / O look, O shine, O let me die and see.
Other than this limited use, a determiner is normally positioned at the beginning of the noun phrase to indicate whether it’s being used in a specific or general sense.
A specific determiner is used when the speaker or writer believes that the listener or reader knows exactly what’s being referred to. The specific determiners are the definite article “the”; the possessives “my,” “your,” “his,” “her,” “its,” “our,” “their,” and “whose”; the demonstratives “this,” “that,” “these,” and “those”; and the interrogative “which.”
On the other hand, a general determiner is used when the speaker or writer is talking about things in general and the listener or reader doesn’t know exactly what’s being referred to. The general determiners are “a,” “an,” “any,” “another,” “other,” and “what.”
