Zero-knowledge proofs vs. "Identification schemes"? (as in Katz--Lindell)
I think I'm qualified to answer this question :-). The approach of going via identification schemes suffices for constructing secure signature schemes. Since this is the aim of this part of the book, it is much simpler than doing full blown zero knowledge. However, if you are interested in looking at sigma protocols and compilation to non-interactive zero knowledge via Fiat-Shamir, then you are going to need the more complex definitions. So, in short, what we do in the book is not more general; on the contrary, it's more specific, but suffices for what we need.