Revealing Hidden Substructures in the $M_{BH}$-$\sigma$ Diagram, and Refining the Bend in the $L$-$\sigma$ Relation

Using 145 early- and late-type galaxies (ETGs and LTGs) with directly-measured super-massive black hole masses, $M_{BH}$, we build upon our previous discoveries that: (i) LTGs, most of which have been alleged to contain a pseudobulge, follow the relation $M_{BH}\propto\,M_{*,sph}^{2.16\pm0.32}$; and (ii) the ETG relation $M_{BH}\propto\,M_{*,sph}^{1.27\pm0.07}$ is an artifact of ETGs with/without disks following parallel $M_{BH}\propto\,M_{*,sph}^{1.9\pm0.2}$ relations which are offset by an order of magnitude in the $M_{BH}$-direction. Here, we searched for substructure in the $M_{BH}$--(central velocity dispersion, $\sigma$) diagram using our recently published, multi-component, galaxy decompositions; investigating divisions based on the presence of a depleted stellar core (major dry-merger), a disk (minor wet/dry-merger, gas accretion), or a bar (evolved unstable disk). The S\'ersic and core-S\'ersic galaxies define two distinct relations: $M_{BH}\propto\sigma^{5.75\pm0.34}$ and $M_{BH}\propto\sigma^{8.64\pm1.10}$, with $\Delta_{rms|BH}=0.55$ and $0.46$~dex, respectively. We also report on the consistency with the slopes and bends in the galaxy luminosity ($L$)--$\sigma$ relation due to S\'ersic and core-S\'ersic ETGs, and LTGs which all have S\'ersic light-profiles. Two distinct relations (superficially) reappear in the $M_{BH}$--$\sigma$ diagram upon separating galaxies with/without a disk (primarily for the ETG sample), while we find no significant offset between barred and non-barred galaxies, nor between galaxies with/without active galactic nuclei. We also address selection biases purported to affect the scaling relations for dynamically-measured $M_{BH}$ samples. Our new, (morphological type)-dependent, $M_{BH}$--$\sigma$ relations more precisely estimate $M_{BH}$ in other galaxies, and hold implications for galaxy/black hole co-evolution theories, and simulations. (Abridged)