(2) This general and powerful framework in (1) provides the rigorous theoretical foundation to derive approximate methods, such as the Linearized spin-mapping approach (Eq. 94 and Eq. 96), which can be used to derive the EOMs proposed in the previous approaches GDTWA (Eq. 98) or Spin-LSC (Eq. 117). It turns out that these two EOMs are identical, just with different dynamical mapping variables. Our work provides the theoretical insights into how seemingly different approaches can be unified into a consistent and rigorous theoretical frame-work.