The well-established correlations between the mass of a galaxy and the properties of its stars are considered to be evidence for mass driving the evolution of the stellar population (SP). However, for early-type galaxies (ETGs), we find that g - i color and stellar metallicity [Z/H] correlate more strongly with gravitational potential Φ than with mass M, whereas SP age correlates best with surface density Σ. Specifically, for our sample of 625 ETGs with integral-field spectroscopy from the Sydney-AAO Multi-object Integral-field Galaxy Survey, compared to correlations with mass, the color-Φ, [Z/H]-Φ, and age-Σ relations show both a smaller scatter and a lower residual trend with galaxy size. For the star formation duration proxy [α/Fe], we find comparable results for trends with Φ and Σ, with both being significantly stronger than the [α/Fe]-M relation. In determining the strength of a trend, we analyze both the overall scatter, and the observational uncertainty on the parameters, in order to compare the intrinsic scatter in each correlation. These results lead us to the following inferences and interpretations: (1) the color-Φ diagram is a more precise tool for determining the developmental stage of the SP than the conventional color-mass diagram; and (2) gravitational potential is the primary regulator of global stellar metallicity, via its relation to the gas escape velocity. Furthermore, we propose the following two mechanisms for the age and [α/Fe] relations with Σ: (a) the age-Σ and [α/Fe]-Σ correlations arise as results of compactness-driven quenching mechanisms; and/or (b) as fossil records of the relation in their disk-dominated progenitors.