  The treatment of convective boundaries during core helium burning is a
fundamental problem in stellar evolution calculations. In Paper~I we showed
that new asteroseismic observations of these stars imply they have either very
large convective cores or semiconvection/partially mixed zones that trap
g-modes. We probe this mixing by inferring the relative lifetimes of asymptotic
giant branch (AGB) and horizontal branch (HB) from $R_2$, the observed ratio of
these stars in recent HST photometry of 48 Galactic globular clusters. Our new
determinations of $R_2$ are more self-consistent than those of previous studies
and our overall calculation of $R_2 = 0.117 \pm 0.005$ is the most
statistically robust now available. We also establish that the luminosity
difference between the HB and the AGB clump is $\Delta
\log{L}_\text{HB}^\text{AGB} = 0.455 \pm 0.012$. Our results accord with
earlier findings that standard models predict a lower $R_2$ than is observed.
We demonstrate that the dominant sources of uncertainty in models are the
prescription for mixing and the stochastic effects that can result from its
numerical treatment. The luminosity probability density functions that we
derive from observations feature a sharp peak near the AGB clump. This
constitutes a strong new argument against core breathing pulses, which broaden
the predicted width of the peak. We conclude that the two mixing schemes that
can match the asteroseismology are capable of matching globular cluster
observations, but only if (i) core breathing pulses are avoided in models with
a semiconvection/partially mixed zone, or (ii) that models with large
convective cores have a particular depth of mixing beneath the Schwarzschild
boundary during subsequent early-AGB `gravonuclear' convection.
