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 RE(3+) cation field strength (CFS) over both short and intermediate length-scales. We explore these dependencies for glasses incorporating the cations La(3+), Y(3+), Lu(3+) and Sc(3+), 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 RE(3+) coordination numbers (Z(RE)) from Z(La) = 6.4 to Z(Sc) = 5.4 in the MD-derived data. However, overall the effects from RE(3+) 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 RE(3+) CFS.