© 2016 Patricio Gaete and José A. Helayël-Neto. Within the framework of the gauge-invariant, but path-dependent, variables formalism, we study the manifestations of vacuum electromagnetic nonlinearities in D=3 models. For this we consider both generalized Born-Infeld and Pagels-Tomboulis-like electrodynamics, as well as Euler-Heisenberg-like electrodynamics. We explicitly show that generalized Born-Infeld and Pagels-Tomboulis-like electrodynamics are equivalent, where the static potential profile contains a long-range (1/r2-type) correction to the Coulomb potential. Interestingly enough, for Euler-Heisenberg-like electrodynamics the interaction energy contains a linear potential, leading to the confinement of static charges.