We investigate the morphology dependence of the Tully-Fisher (TF) relation, and the expansion of the relation into a three-dimensional manifold defined by luminosity, total circular velocity and a third dynamical parameter, to fully characterize spiral galaxies across all morphological types. We use a full semi-analytic hierarchical model (based on Croton et al.), built on cosmological simulations of structure formation, to model galaxy evolution and build the theoretical TF relation. With this tool, we analyse a unique data set of galaxies for which we cross-match luminosity with total circular velocity and central velocity dispersion. We provide a theoretical framework to calculate such measurable quantities from hierarchical semi-analytic models. We establish the morphology dependence of the TF relation in both model and data. We analyse the dynamical properties of the model galaxies and determine that the parameter σ/VC, i.e. the ratio between random and total motions defined by velocity dispersion and circular velocity, accurately characterizes the varying slope of the TF relation for different model galaxy types. We apply these dynamical cuts to the observed galaxies and find indeed that such selection produces a differential slope of the TF relation. The TF slope in different ranges of σ/VC is consistent with that for the traditional photometric classification in Sa, Sb and Sc. We conclude that σ/VC is a good parameter to classify galaxy type, and we argue that such classification based on dynamics more closely mirrors the physical properties of the observed galaxies, compared to visual (photometric) classification. We also argue that dynamical classification is useful for samples where eye inspection is not reliable or impractical. We conclude that σ/VC is a suitable parameter to characterize the hierarchical assembly history that determines the disc-to-bulge ratio, and to expand the TF relation into a three-dimensional manifold, defined by luminosity, circular velocity and σ/VC.