Compared to the Monte Carlo wavefunction methods which randomly collapse the wavefunction onto single states, the L-MFE approach only adds on random phases to the expansion coefficients, providing a more stable dynamics which can be incorporated with any mixed quantum-classical, semiclassical, or wavepacket based approaches. Compared to the Ehrenfest+R method the L-MFE method uses the same procedure to decay the magnitude of the quantum expansion coefficients, but a different choice of the random phase distribution, such that the exact Lindblad dynamics can be recovered, whereas Ehrenfest+R cannot exactly recover Lindblad dynamics. The derivation procedure of L-MFE also does not assume any relation between the energy gap of the two states versus the decay rate, whereas Ehrenfest+R assumes a particular parameter regime where the energy gap is much larger than the decay rate. Our theoretical analysis further provides valuable insights and mathematical justification for the relationship between the dynamics of the reduced density matrix and the dynamics of the ensemble of pure states. Through these careful analyses, we discovered an easy fix of the Ehrenfest+R method with the correct "+R" decay rate, which can be used to fix Ehrenfest+R dynamics to exactly reproduce Lindblad dynamics. Throughout the theoretical development in this work, we demonstrated the importance of including a carefully chosen random phase on both the coherences as well as the diagonal populations of the dynamics. Using numerical simulations, we demonstrated that the L-MFE method is equivalent to Lindblad dynamics for a variety of complicated dynamical scenarios when nuclear DOFs are not present, including scenarios where previous approaches (such as Ehrenfest+R ) do not match Lindblad dynamics. We further demonstrated that when including nuclei in Ehrenfest dynamics, the L-MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach.