A while ago I was looking for a distribution over discrete variables where I could control both the mean and the variance separately. As you clearly explained, the negative binomial only handles the overdispersed case, the Poisson distribution handles the case when mean and variance are equal and the binomial handles the underdispersed case.

I finally ran into this distribution: Conway-Maxwell-Poisson. This distribution allows one to control mean and variance directly. Interesting aside: it is an exponential family family distribution (which doesn’t make it easier to work with though).