This paper studies pseudo-bulges (P-bulges) and classical bulges (C-bulges) in Sloan Digital Sky Survey central galaxies using the new bulge indicator $DeltaSigma_1$, which measures relative central stellar-mass surface density within 1 kpc. We compare $DeltaSigma_1$ to the established bulge-type indicator $Deltalanglemu_erangle$ from Gadotti (2009) and show that classifying by $DeltaSigma_1$ agrees well with $Deltalanglemu_erangle$. $DeltaSigma_1$ requires no bulge-disk decomposition and can be measured on SDSS images out to $z = 0.07$. Bulge types using it are mapped onto twenty different structural and stellar-population properties for 12,000 SDSS central galaxies with masses 10.0 < log $M_*$/$M_{odot}$ < 10.4. New trends emerge from this large sample. Structural parameters show fairly linear log-log relations vs. $DeltaSigma_1$ and $Deltalanglemu_erangle$ with only moderate scatter, while stellar-population parameters show a highly non-linear elbow in which specific star-formation rate remains roughly flat with increasing central density and then falls rapidly at the elbow, where galaxies begin to quench. P-bulges occupy the low-density end of the horizontal arm of the elbow and are universally star-forming, while C-bulges occupy the elbow and the vertical branch and exhibit a wide range of star-formation rates at fixed density. The non-linear relation between central density and star-formation rate has been seen before, but this mapping onto bulge class is new. The wide range of star-formation rates in C-bulges helps to explain why bulge classifications using different parameters have sometimes disagreed in the past. The elbow-shaped relation between density and stellar indices suggests that central structure and stellar-populations evolve at different rates as galaxies begin to quench.
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