자신 can be used to talk about the self moreso than 자기 can.
내가 자신을 믿어요 = I believe in myself.
이 영화는 제 자신에 대한 것이에요 = The movie is about me.
그가 자신에게 자주 말해 = He talks to himself a lot.
자기 is more like using a pronoun when you've already mentioned the person.
아빠가 자기 건강에 대해 걱정해요 = My dad is concerned about his health.
Hmm, is 자기의 책을 in this context only reading as "her book" because it's established that we're talking about a woman, ergo SELF+Possessive+ Book = Self's book = (Her)self's book = Her book?
Is that the logic here or is it something else?
Yes, the 자기 is basically a placeholder here or to prevent redundancies. It would be very weird to say "the woman is reading the woman's (own) book". Does that mean one woman is reading another woman's book? Or is she reading her book? Confusion. So the idea is that 자기의 would act as "her/his" in this case.
Again, you wouldn't say "Danielle is reading Danielle's (own) book" as that would cause confusion. You would, however, say "Danielle is reading her (own) book".
자기의 basically means "one's own". Her own, his own, it's own, my own etc.
Your translation, that is 'her book' is a tad bit off, because it should also contain the word 'own'.
Just to clarify after reading and trying to understand the other comments:
Would 자신 be used like a noun that means "self" (making it most suitable when used with a pronoun to become "myself", "himself", and "herself"), while 자기 would be used like a pronoun only after the subject noun it refers to has been stated (basically to mean "his/her" in reference to the subject noun)?
Sorry if that sounds confusing lol, I'll gladly further explain what I mean if asked.