Black-Box Zero Knowledge vs Non-Black-Box Zero Knowledge I'm confused about the definitions of black-box zero knowledge and non-black-box zero knowledge. I have searched and found explanations but still a bit confused about it. From the context, their definition (with auxiliary input) of black-box zero knowledge proof system between the prover $P$ and (malicious) verifier $V^*$ can be described as follows: 

$\{\langle P(\omega), V^*(z)(x)\rangle\}_{(x,\omega) \in R, z \in \{0,1\}^{p(|x|)}}$ 

and $\{S^{V^*(z,x,\cdot)}(x)\}_{(x,\omega) \in R, z \in \{0,1\}^{p(|x|)}}$ 

are (perfectly/statistically/computationally) indistinguishable, Where $S$ is a simulator which has access to a oracle function $V^*(z,x,\cdot)$

In my opinion, the only difference between the definition of non-black-box zero knowledge and the definition of black-box zero knowledge is that non-black-box zero knowledge allows that for all $V^*$ there exists unbounded simulator $S$, is this correct?
The difference is to do with the quantifiers on the simulator. Non black-box zero-knowledge says that for all verifiers, there exists a simulator that can simulate a proof of all statements (i.e. there can be one simulator per verifier). Black-box zero knowledge says that there exists a simulator that can simulate a proof of all statements over all verifiers (i.e. there is one simulator total). Therefore, black-box zero-knowledge is the stronger definition of security.

Reference: Lysyanskaya, 2002 (page 31) https://dspace.mit.edu/handle/1721.1/29271