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Understanding the variation of thermal conduction in disordered oxides is important for applications related to energy saving and electronics but currently lacks fundamental insight into the phonon propagation mechanisms. In this Letter, we report a strong correlation between thermal conductivity and the speed of sound within two families of modified oxide glass formers, in agreement with phonon-gas-theory predictions for crystalline materials. Based on calculations of diffuson-mediated heat transport, we then show that thermal conductivity in borate glasses is dominated by diffusive vibrational modes, while both diffusive and propagative modes contribute to heat conduction in silicate glasses. This fundamental difference in the heat propagation mechanism originates from differences in the phonon mean free path of low-frequency modes caused by the inherent variation of atomic rigidity between silicate and borate glasses.