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In this paper, the self-consistent PMMCS force fields (FFs) [Pedone et al., J. Phys. Chem. B 110, 11780 (2006)] widely used for the simulation of a large variety of silicates, aluminosilicate and phosphate crystals, and multicomponent oxide glasses have been revised and improved by the inclusion of two types of three-body interactions acting between T-O-T bridges ($T=\mathrm{Si}$ and P) and network former-network former repulsive interactions. The FFs named Bertani--Menziani--Pedone (BMP)-harm and BMP-shrm better reproduce the T-O-T bond angle distributions (BADs) and network former-oxygen distances. Consequently, the prediction of ${Q}^{n}$ distributions (Q stands for quaternary species, and $n$ is the number of bridging oxygens around it), neutron total distribution functions, solid-state nuclear magnetic resonance spectra of spin active nuclei ($^{29}\mathrm{Si}$, $^{17}\mathrm{O}$, $^{31}\mathrm{P}$, $^{27}\mathrm{Al}$), and the density have also been hugely improved with respect to the previous version of our FF. These results also highlight the strong correlation between the T-O-T BADs and the other short and intermediate structural properties in oxide glasses, which have been largely neglected in the past. In addition to the improvement of the structure, the FF has been revealed to reproduce well the ionic conductivity in mixed alkali aluminosilicate glasses and the elastic properties. The systematic comparison with other interatomic potential models, including the polarizable core-shell model, carried out in this paper showed that our potential model is more balanced and effective for simulating a vast family of crystalline and amorphous oxide-based systems.