Is Using Digital Signatures to prove identity a zero knowledge proof? Suppose Alice publishes a book with a public key in it, and later wants to prove that she wrote the book. She could sign challenge messages with her private key, and others could verify those signed messages. Alice is proving she knows a private key without revealing anything about it.

Is this an example of an interactive zero knowledge proof? 

Or does this example somehow not apply to zero knowledge proofs?
This is not zero knowledge. In particular, you give away information in the form of signatures on challenges. This is something that the verifier doesn't have and so it is something that is "learned". 

This can be meaningful for two reasons. Let's say that I want to prove to YOU that I wrote the book, but I don't want you to be able to convince anyone else that you interacted with the person who wrote the book. With this protocol, it's possible to prove that you interacted with the book author, but this isn't possible with real ZK. Another issue is that the challenge may actually be generated maliciously so that it provides a meaningful signature.