Using a mixed Wigner/S-W formalism, we derive a general expression of the time-correlation function, where the Wigner representation is used for the nuclear DOFs, and the S-W transform is applied to the generalized spin matrices associated to the electronic DOFs. We obtain the expression of the exact quantum Liouvillian in this formalism. Further making a linearization approximation, we obtain a set of EOMs that describe the coupled dynamics between the electronic and nuclear DOFs. We further connect EOMs with different mapping variables, including the spin coherence state variables, generalized Bloch spherical coordinates, as well as the MMST mapping variables. We formally establish the equivalence of these EOMs with different mapping variables. We also connect a variety of previously developed methods with the current formalism in the language of the SU (N ) mapping formalism. Finally, we perform numerical simulations to assess the accuracy of the generalized spin mapping approach under the linearization approximation. We compute the population dynamics of systems with multiple electronic states coupled to the nuclear DOFs, including a condensed phase spin-boson model system, two conical intersection models, and an anharmonic three-state Morse model for photo-dissociation dynamics. The current formalism provides an excellent agreement compared to the numerically exact results, and a significant improvement compared to the Ehrenfest dynamics or LSC-IVR which is based on the MMST mapping formalism. Interestingly, the current formalism produces very similar numerical results compared to two recently developed approaches, Spin-LSC and GDTWA, both of which are based on the SU (N ) mapping formalism.