adjectives rule

They're their own special class of word, but they follow more adjective rules than noun rules, so you'll see them called all kinds of things

Does that make the present participle (progressive) forms of verbs like "writing" adjectives too, because they're used like adjectives rather than like verbs ("I am tired", "I am writing", not "I writing")? Does it make gerunds (unfortunately identical with present participles in English) nouns because they're used like nouns ("I like cats", "I like writing")?

They're each their own class of word, neither strictly verb (despite being verb forms) nor adjective / noun (despite being used similarly) and I think we should treat them as such.

I learned them in line with the declension of pronouns as analogues of nouns (which is generally less significant in English): I, my, me; you, your, you; he, his, him; she, her, her; it, its, it and so on. In some languages they have adjective-like declensions, but I think calling them adjectives is imprecise, because they're more than that: They generally don't make much sense without something they're substituted for.

For first and second person, that isn't immediately obvious, but for third person, you can probably see what I mean: "Her car is rusty. Its tires are flat."
We can infer that "Its" refers to the car, but "Her" doesn't tell us anything useful if we don't know who that "She" is. That is a trait of pronouns: They're substituted pro nomine.

I can say "tired cat" and it is a semantically complete. "Her car" isn't.

Hence: I concede that they're not purely pronouns, but rather pronouns with characteristics of adjectives. Their semantic intention is still to reference some other noun in the context, so I don't think it's fair to call them adjectives either.