We analyze the class of nonlinear electrodynamics minimally coupled to gravitation supporting asymptotically flat non-Schwarzschild-like elementary solutions. The Lagrangian densities governing the dynamics of these models in flat space are defined and fully characterized as a subclass of the set of functions of the two standard field invariants, restricted by requirements of regularity, parity invariance and positivity of the energy, which are necessary conditions for the theories to be physically admissible. Such requirements allow for a complete characterization and classification of the geometrical structures of the elementary solutions for the corresponding gravity-coupled models. In particular, an immediate consequence of the requirement of positivity of the energy is the asymptotic flatness of gravitating elementary solutions for any admissible model. The present analysis, together with the (already published) one concerning the full class of admissible gravitating nonlinear electrodynamics supporting asymptotically flat Schwarzschild-like elementary solutions, completes and exhausts the study of the gravitating pointlike charge problem for this kind of models.