In the linear regression model $Y= X\beta+u$ , when $X$ is endogenous , the OLS estimator will be biased. If we can find an instrumental variable $Z$ which is correlated with $X$, but uncorrelated with the error term, we can get the unbiased coefficient $\beta$. This requirement condition is also what I learned from my econometric classes. For example in Wooldridge book Introductory Econometrics Modern Approach (2018) chapter 15, the condition is $ cov(z, u)=0$,$cov(z, x)\neq0$.
However, when I check some material on line, it seems there is another type of requirements. The requirements are listed below:
(i) $Z$ is related to $X$,causally or not causally;
(ii) $Z$ affects the outcome variable $Y$ only through $X$ ($Z$ does not have a direct influence on $Y$ which is referred to as the exclusion restriction);
(iii) There is no confounding for the effect of $Z$ on $Y$
The same condition is used in Robins (2020) Causal Inference book chapter 16.
The key difference is in the second set condition (iii) $Z$ and $Y$ don't share common confound factors, which is not required in first set conditions. From my understanding, if we have the condition (ii), it means $Z$ can only affect $Y$ through $X$. In this case, we should have $ cov(z, u)=0$. Why do we bother to the condition (iii). If condition (iii) is violated, does it mean $Z$ could be related to $Y$ through other channel rather than $X$.
Do I miss something here?