Sets of self-consistent oxygen-rare earth (RE = La, Y, Lu, Sc) interatomic potential parameters are derived using a force-matching procedure and utilized in molecular dynamics (MD) simulations for exploring the structures of RE₂O₃–Al₂O₃–SiO₂ glasses that feature a fixed molar ratio n_Al/n_Si = 1 but variable RE contents. The structures of RE aluminosilicate (AS) glasses depend markedly on the RE³⁺ cation field strength (CFS) over both short and intermediate length-scales. We explore these dependencies for glasses incorporating the cations La³⁺, Y³⁺, Lu³⁺ and Sc³⁺, whose CFSs increase due to the concomitant shrinkage of the ionic radii: R_La > R_Y > R_Lu > R_Sc. This trend is mirrored in decreasing average RE³⁺ coordination numbers (_RE) from _La = 6.4 to _Sc = 5.4 in the MD-derived data. However, overall the effects from RE³⁺ CFS elevations on the local glass structures are most pronounced in the O and {Al^[4], Al^[5], Al^[6]} speciations. The former display minor but growing populations of O^[0] (“free oxygen ion”) and O^[3] (“oxygen tricluster”) moieties. The abundance of AlO₅ polyhedra increases significantly from ≈10% in La-based glasses to ≈30% in their Sc counterparts at the expense of the overall dominating AlO₄ tetrahedra, whereas the amounts of AlO₆ groups remain <5% throughout. We also discuss the Si^[4]/Al^[p] (p = 4, 5, 6) intermixing and the nature of their oxygen bridges, where the degree of edge-sharing increases together with the RE³⁺ CFS.