After decades of searching, cosmological time dilation was recently identified in the timescale of variability seen in distant quasars. Here, we expand on the previous analysis to disentangle this cosmological signal from the influence of the properties of the source population, specifically the quasar bolometric luminosity and the rest-frame emission wavelength at which the variability was observed. Furthermore, we consider the potential influence of the evolution of the quasar population over cosmic time. We find that a significant intrinsic scatter of 0.288 +- 0.021 dex in the variability timescales, which was not considered in the previous analysis, is favoured by the data. This slightly increases the uncertainty in the results. However, the expected cosmological dependence of the variability timescales is confirmed to be robust to changes in the underlying assumptions. We find that the variability timescales increase smoothly with both wavelength and bolometric luminosity, and that black hole mass has no effect on the variability timescale once rest wavelength and bolometric luminosity are accounted for. Moreover, if the standard cosmological model is correct, governed by relativistic expansion, we also find very little cosmological evolution in the intrinsic variability timescales of distant quasars.
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