how does omitted-variable bias violate exogeneity

I understand that when omitted-variable bias occurs the coefficient estimated for some regressors is the sum of the direct effect and indirect effect through the omitted-variable. What I fail to see is how this causes exogeneity to fail. Let's say the true relationship is given by $$y= c+ ax+bz+error$$ where the error term fulfills all the assumptions and $$x$$ and $$z$$ are correlated. Now if we run a regression with only a constant and $$x$$ we will get some result $$y=f+gx+error$$. Now the coefficient $$g$$ will be higher or lower than $$a$$ depending on the direction of the bias. However wouldn't the error term in this second equation still fulfill exogeneity?

edit: To clarify my question consider the following. Let's say $$y= c+ ax+bz+error$$ (where the error term fulfills all the assumptions) descibes reality. If we have $$z=vx$$ than $$y=h+mx+error$$ will also describe reality. When reading about OVB I have seen statemets like "if the true modell is $$y= c+ ax+bz+error$$ and x and z are correlated than regressing only on x will overestimate(or underestimate) the true coefficient of $$x$$". But this seems to be trying to establish causality? If we are only talking about descriptive models than there could be many different correct coefficients for $$x$$ depending on the variables included in the model couldn't there?

Exogeneity won't hold if the correlation between z and x is nonzero and $$b\not=0$$ because then the error will be correlated to x in model 2 with the omitted z (which is now part of the error). If the true value of b is zero then you would also still have exogeneity. Deriving the estimator of a in both cases might help you see this.