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 ∝ M 2.16±0.32 * ,sph ; and (ii) the ETG relation M BH ∝ M 1.27±0.07 * ,sph is an artifact of ETGs with/without disks following parallel M BH ∝ M 1.9±0.2 * ,sph 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, σ) diagram using our recently published, multicomponent, 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 Sérsic and core-Sérsic galaxies define two distinct relations: M BH ∝ σ 5.75±0.34 and M BH ∝ σ 8.64±1.10 , with ∆ 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)-σ relation due to Sérsic and core-Sérsic ETGs, and LTGs which all have Sérsic light-profiles. Two distinct relations (superficially) reappear in the M BHσ 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 -σ relations more precisely estimate M BH in other galaxies, and hold implications for galaxy/black hole co-evolution theories, simulations, feedback, the pursuit of a black hole fundamental plane, and calibration of virial f -factors for reverberation-mapping.