Defining the (Black Hole)-Spheroid Connection with the Discovery of
  Morphology-Dependent Substructure in the $M_{\rm BH}$--$\rm n_{sph}$ and
  $M_{\rm BH}$--$\rm R_{e, sph}$ Diagrams: New Tests for Advanced Theories and
  Realistic Simulations
  For 123 local galaxies with directly-measured black hole masses ($M_{\rm
BH}$), we provide the host spheroid's S\'ersic index ($\rm n_{sph}$), effective
half-light radius ($\rm R_{e,sph}$), and effective surface brightness
($\mu_e$), obtained from careful multi-component decompositions, and we use
these to derive the morphology-dependent $M_{\rm BH}$--$\rm n_{sph}$ and
$M_{\rm BH}$--$\rm R_{e,sph}$ relations. We additionally present the
morphology-dependent $M_{\rm *,sph}$--$\rm n_{sph}$ and $M_{\rm *,sph}$--$\rm
R_{e,sph}$ relations. We explored differences due to: early-type galaxies
(ETGs) versus late-type galaxies (LTGs); S\'ersic versus core-S\'ersic
galaxies; barred versus non-barred galaxies; and galaxies with and without a
stellar disk. We detect two different $M_{\rm BH}$--$\rm n_{sph}$ relations due
to ETGs and LTGs with power-law slopes $3.95\pm0.34$ and $2.85\pm 0.31$. We
additionally quantified the correlation between $M_{\rm BH}$ and the spheroid's
central concentration index, which varies monotonically with the S\'ersic
index. Furthermore, we observe a single, near-linear $M_{\rm *,sph}$--$\rm
R_{e,sph}^{1.08\pm 0.04}$ relation for ETGs and LTGs, which encompasses both
classical and alleged pseudobulges. In contrast, ETGs and LTGs define two
distinct $M_{\rm BH}$--$\rm R_{e,sph}$ relations with $\Delta_{\rm
rms|BH}\sim\rm 0.60~dex$ (cf.\ $\sim$0.51~dex for the $M_{\rm BH}$--$\sigma$
relation and $\sim$0.58~dex for the $M_{\rm BH}$--$M_{\rm *,sph}$ relation),
and the ETGs alone define two steeper $M_{\rm BH}$--$\rm R_{e,sph}$ relations,
offset by $\sim$1~dex in the $\log M_{\rm BH}$-direction, depending on whether
they have a disk or not and explaining their similar offset in the $M_{\rm
BH}$--$M_{\rm *,sph}$ diagram. This trend holds using $10 \%$, $50 \%$, or $90
\%$ radii.(Abridged)
