Most modern papers on zero-knowledge proofs (for example, BBB+16) recycle essentially the same list of boilerplate definitions, concerning soundness (formulated using extraction or some form of "emulation") and zero-knowledge (involving simulation).
Yet Katz and Lindell's incredible book does not use these definitions—instead speaks only of an Identification scheme (see [KL15, p. 452]), which is either secure (against a passive attack), or isn't (i.e., see Def. 12.8). For example, the Schnorr protocol is formulated only as an identification scheme (Fig. 12.2). Moreover, both its perfect zero knowledge and 2-soundness are implicitly contained in its proof of security (Thm. 12.11).
My question is: should I view this simpler Identification scheme-based definition as "the future"? Can we get by with definitions of this sort, i.e., can this definition capture everything we "care about" in zero-knowledge proofs? What are the reasons for keeping the more traditional definitions around, if any?
Moreover, has an Identification scheme-based definition been formulated for more general settings, where the "statement" of the relevant NP language has a complicated structure (is no longer uniformly random, maybe selected by the adversary, etc.)? Where can I find these more general formulations?