The (Black Hole Mass)-(Spheroid Stellar Density) Relations: $M_{\rm
  BH}$--$\mu$ (and $M_{\rm BH}$--$\Sigma$) and $M_{\rm BH}$--$\rho$
  This paper is the fourth in a series presenting (galaxy morphology, and thus
galaxy formation)-dependent black hole mass, $M_{\rm BH}$, scaling relations.
We have used a sample of 119 galaxies with directly-measured $M_{\rm BH}$ and
host spheroid parameters obtained from multi-component decomposition of,
primarily, $3.6\,\mu$m Spitzer images. Here, we investigate the correlations
between $M_{\rm BH}$ and the projected luminosity density $\mu$, the projected
stellar mass density $\Sigma$, and the deprojected (internal) stellar mass
density $\rho$, for various spheroid radii. We discover the predicted $M_{\rm
BH}$--$\mu_{\rm 0,sph}$ relation and present the first $M_{\rm BH}$--$\mu_{\rm
e, sph}$ and $M_{\rm BH}$--$\rho_{\rm e,int, sph}$ diagrams displaying slightly
different (possibly curved) trends for early- and late-type galaxies (ETGs and
LTGs) and an offset between ETGs with (fast-rotators, ES/S0) and without
(slow-rotators, E) a disk. The scatter about various $M_{\rm
BH}$--$\langle\Sigma\rangle_{\rm R,sph}$ (and $\langle\rho\rangle_{\rm r,sph}$)
relations is shown to systematically decrease as the enclosing aperture (and
volume) increases, dropping from 0.69~dex when using the spheroid
\enquote{compactness}, $\langle\Sigma\rangle_{\rm 1kpc,sph}$, to 0.59~dex when
using $\langle\Sigma\rangle_{\rm 5kpc,sph}$. We also reveal that $M_{\rm BH}$
correlates with the internal density, $\rho_{\rm soi,sph}$, at the BH's
sphere-of-influence radius, such that core-S\'ersic (high S\'ersic index, $n$)
and (low-$n$) S\'ersic galaxies define different relations with total rms
scatters 0.21~dex and 0.77~dex, respectively.(Abridged)
