Why is the link function a function of the mean and not the linear predictor? Generalized linear models are formulated so that the link function $g$ of the mean $\mu$ of a random variable $Y$ is equal to the linear predictor $\eta=x'\beta$, i.e.
\begin{align}
g(\mu)=\eta.
\end{align}
My question is, why don't we call $h=g^{-1}$ the link function and write $\mu=h(\eta)$. That way, if $h$ is not one-to-one, there is still a distribution on the response variables but the $\beta$ are not identifiable. Isn't it problematic that for certain $g$, there is no well-defined model? It also is more intuitive to think of the mean as a function of the parameters, rather than the reverse. Is there a conceptual advantage to defining $g$ in this way? Is it mathematically necessary for some reason I'm not seeing? 
 A: First of all, the notation is not used consistently. For example, Nelder and  Wedderburn (1972) write about linear predictor $Y$ and a linking function "$\theta = f(Y)$ connecting the parameter $\theta$ of the distribution of $z$ with the $Y$'s of the linear model" (p. 372). On another hand, McCullaugh and Nelder (1983) use the link function $\eta_i = g(\mu_i)$ (p. 27). So both notations were used in the classical literature on GLM's.
Basically it doesn't matter if you use $f = g^{-1}$ since what you need is a one-to-one mapping that works in both direction (through it's inverse). All the common link functions have this property. Nobody said that $g$ can be any function...
As about "intuitiveness", notice that if you waned to use linear regression for a count data, then you would usually transform the outcome using a log transformation and then fit a linear regression to it, so in some cases transforming the outcome is also "intuitive".
Moreover, I don't find anything more intuitive in considering transformed mean as a function of linear predictor vs mean as a function of transformed linear predictor, they are the same.

Nelder, J. and Wedderburn, R. (1972). Generalized Linear Models.
  Journal of the Royal Statistical Society. Series A (General).
  Blackwell Publishing. 135 (3): 370–384.
McCullagh, P. and Nelder, J. (1989). Generalized Linear Models, Second
  Edition. Boca Raton: Chapman and Hall/CRC.

