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For decades transition-metal oxides have generated a huge interest due to the multitude of physical phenomena they exhibit. In this class of materials, the rare-earth nickelates, $R$NiO$_3$, stand out for their rich phase diagram stemming from complex couplings between the lattice, electronic and magnetic degrees of freedom. Here, we present a first-principles study of the low-temperature phase for two members of the $R$NiO$_3$ series, with $R=$ Pr, Y. We employ density-functional theory with Hubbard corrections accounting not only for the on-site localizing interactions among the Ni--$3d$ electrons ($U$), but also the inter-site hybridization effects between the transition-metals and the ligands ($V$). All the \textit{U} and \textit{V} parameters are calculated from first-principles using density-functional perturbation theory, resulting in a fully \emph{ab initio} methodology. Our simulations show that the inclusion of the inter-site interaction parameters $V$ is necessary to simultaneously capture the features well-established by experimental characterizations of the low-temperature state: insulating character, antiferromagnetism and bond disproportionation. On the contrary, for some magnetic orderings the inclusion of on-site interaction parameters $U$ alone completely suppresses the breathing distortion occurring in the low-temperature phase and produces an erroneous electronic state with a vanishing band gap. In addition -- only when both the \textit{U} and \textit{V} are considered -- we predict a polar phase with a magnetization-dependent electric polarization, supporting very recent experimental observations that suggest a possible occurrence of type-II multiferroicity for these materials.