In order to investigate certain classes of natural exponential families (NEFs), we have suggested to modify their cumulant functions to other convex functions of their Laplace transforms. One of the main objectives is to get an easier function of the characteristic second-moments. Illustrations are carried out for geometric sums of NEFs, namely geometric dispersion models characterized by their v-functions, and for factorial or discrete dispersion models characterized by their dispersion functions. Two particular situations are mainly exhibited from tilting exponentials of $\alpha$-stable distributions, so-called Tweedie models or power variance functions, which provide geometric-Tweedie and Poisson-Tweedie models having power v-functions and power dispersion functions, respectively. They are also considered for asymptotic distributions in their respective families of models, leading to analog of some classical limit distributions such Central Limit Theorem, Poisson's convergence and R\'enyi's theorem. Extended problems will be addressed through the combined models of geometric-Poisson-Tweedie, also referred to as Poisson-geometric-Tweedie.