We examined aluminosilicate glasses containing a variety of network modifying to intermediate cations (Li, La, Sc, and Fe), quenched from melts at 1 atm to 8 GPa, to further investigate the role of cation field strength in Al coordination changes and densification. 27Al Nuclear Magnetic Resonance Spectroscopy (NMR) reveals that the mean Al coordination increases with increasing pressure in the Li-containing glasses, which can be explained by a linear dependence of fractional change in Al coordination number on cation field strengths in similar K-, Na-, and Ca-containing aluminosilicate glasses (K < Na < Li < Ca). Measured recovered densities follow a similar linear trend. In contrast, the La-containing glasses have significantly lower mean Al coordination numbers at given pressures than the cation field strength of La and glass density would predict. La L3 X-ray absorption fine structure (XAFS) spectroscopy results indicate a significant increase with pressure in average La-O bond distances, suggesting that La and Al may be "competing" for higher coordinated sites and hence that both play a significant role in the densification of these glasses, especially in the lower pressure range. However, in Na aluminosilicate glasses with small amounts of Sc, 45Sc NMR reveals only modest Sc coordination changes, which do not seem to significantly affect the mean Al coordination values. For a Li aluminosilicate glass, 17O MAS and multiple quantum magic angle spinning (3QMAS) NMR data are consistent with generation of more highly coordinated Al at the expense of non-bridging oxygen (NBO), whereas La aluminosilicate glasses have roughly constant O environments, even up to 8 GPa. Finally, we demonstrate that useful 23Na and 27Al MAS NMR spectra can be collected for Ca-Na aluminosilicate glasses containing up to 5 wt.% Fe oxide. We discuss the types of structural changes that may accompany density increases with pressure and how these structural changes are affected by the presence of different cations.