Can distributions that are in the exponential family, but not the natural exponential family, be formed as GLM? The lognormal and beta distributions are in the exponential family but not the natural exponential family. 
Generalized Linear Models are often advertised as being models for response variables that are assumed to follow an exponential family (see wikipedia).
However, I know beta regression is not a GLM (see Why Beta/Dirichlet Regression are not considered Generalized Linear Models?) and lognormal regression is also not a GLM (How to specify a lognormal distribution in the glm family argument in R?).
What's the discrepancy here? Is there some way to model response variables as coming from a lognormal or beta distribution that can be formed as a GLM that is somehow different from "beta regression" or "lognormal regression"? Or is it just sloppy language and a GLM assumes a natural exponential family.
 A: To be in the McCullagh and Nelder framework you need the distribution to be in the exponential dispersion family, an extension of the natural exponential family but not equivalent to the full exponential family.
One of the things you need: in the exponential family you have a term $\eta (\theta )\cdot T(x)$ in the exponent, for GLMs $T$ must be the identity. This excludes the lognormal and beta (though this isn't the only issue).
[This much of the answer is covered in answers at your first link]
You could use lognormal in glms after a transformation of the response (by $T$, so log in the case of lognormal). However, then a GLM model is not for the mean of the response but for the mean of the transformed response. If you're careful about translating the parameter estimates back to the original scale you can then produce an MLE of the mean of the original (I have done this for inverse gamma models with log link, for example - inverting the data, getting MLEs for parameters, and then using those parameter estimates in the inverse gamma to obtain expressions for the mean (this treats the shape parameter separately; you can estimate the mean of the gamma in a shape-mean parameterization without identifying the shape, then estimate the MLE for the shape given the MLE for $\mu$; the R package MASS makes this easy).
It's not immediately clear that this sort of 'trick' would necessarily work for the beta (after a logit transform) but I haven't tried to work through it to see - I expect it is not possible in that case, in that I don't think it will be doable the way it can be done for the gamma. However, perhaps it might just be possible to use a similar approach as was done for the negative binomial (again, see MASS).
