You have a page on possessive adjectives: http://www.grammar-monster.com/glossary/possessive_adjectives.htm

Some of the information under the heading "Possessive Adjectives are a Type of Pronoun" is incorrect:

You said that possessive adjectives replace nouns.
This is incorrect. Possessive adjectives show ownership. They come before nouns (e.g., HER car); they don't replace nouns.

You said that possessive adjectives are also called possessive pronouns.
Incorrect. They are two different things entirely.
POSSESSIVE ADJECTIVE: This is HER car.
POSSESSIVE PRONOUN: The car is HERS.

Possessive adjectives describe the noun.
Possessive pronouns replace the noun.

Sometimes they have the same form so you have to look at each sentence individually to see whether it's a possessive adjective or a possessive pronoun.
POSSESSIVE ADJECTIVE: This is HIS car.
POSSESSIVE PRONOUN: The car is HIS.

As it stands now, your explanation doesn't work.

PM: The so-called "possessive adjectives", "my, your, his, her, its, our, your" are not adjectives at all; they belong to the word category (part of speech) of pronouns, which is a sub-class of noun. They typically function as determiners in noun phrases, as in "I’ve lost my key"; "Ed has just sold his house" and so on.

There are two kinds of possessive pronoun: "dependent" and "independent". The dependent kind ("my, your" etc.) requires a following noun (hence the name dependent) as shown in the above examples. Unlike the dependent kind, the independent kind "mine, yours, his, hers, its, ours, yours and theirs" can stand alone as in "This beer is yours, but that one is mine" hence the name "independent". But note that both kinds belong to the category of pronouns, though their functions in the sentence differ somewhat.

PM: The dependent kind of possessive pronoun ("my, your" etc.) does occur before nouns, it's true, and they do typically indicate possession (ownership). But they do in general "replace" nouns, which is why they are called pronouns; compare: A: "I saw Ed’s new car yesterday" ~ B: "I saw his new car, too", where in the latter "his" is used instead of "David’s" to avoid unnecessary repetition of the noun and is said to be "anaphoric" to the noun "David’s", called the antecedent.

PM: Traditional grammar calls them possessive adjectives and possessive pronouns, but in modern (and scholarly) grammar they are simply called possessive pronouns which may be of the dependent or independent kind. But both forms of the personal pronouns, i.e. "her" and "hers" are just as much pronouns as "she" and "her" are. See also my comments above.

PM: Yes, the 3rd person singular dependent and independent possessive pronouns have the same form, "his" and "its". The others have distinctly different inflectional forms. Your two examples exhibit possessive pronouns of the independent and dependent kind respectively.

PM: In general, improvements in the way we analyse the language have resulted in numerous changes of terminology. Those items that traditional grammar calls "possessive adjectives" are now called "possessive pronouns" and their function is identified as "determiner". In scholarly grammar, even the term "possessive" has been largely replaced by the term "genitive". Regrettably, some Internet grammar sites (including GM) have not caught up yet, and even some dictionaries are still lagging behind.

Does that address your concerns?

PaulM

James: see
http://www.softschools.com/examples/grammar/possessive_adjectives_versus_possessive_pronouns_examples/82/

Yes - it's all there: .....possessive_pronouns_examples/82/

Warning - Do not Trust The Above Website! - PM

The above link takes the reader to a "Mickey Mouse" website for raw beginners based on nonsense out-of-date grammar. Regrettably, it's not alone: there are quite a few others like it.

No one who takes grammar seriously calls the "his" of "his car", or the "their" of "their house" possessive adjectives. They are possessive pronouns functioning as possessive determiners. UCL (University College London) is a genuinely useful resource:

http://www.ucl.ac.uk/internet-grammar/determin/pronoun.htm

As is the Cambridge dictionary:

http://dictionary.cambridge.org/grammar/british-grammar/pronouns/pronouns-possessive-my-mine-your-yours-etc.

And the Oxford dictionary too:

https://en.oxforddictionaries.com/definition/his.

Both dictionaries call such items "possessive determiners", which is actually their function - not their word category - but at least they don't make the child-like mistake of calling them possessive adjectives!

PaulM