Two lattice Boltzmann method (LBM) formulations are possible to account for the effect of the magnetic field on the velocity field in magnetohydrodynamic (MHD) flows. In the body-force formulation (BFF), the magnetic field effects manifest as an external acceleration. In the extended equilibrium formulation (EEF), the effect appears through a modified equilibrium distribution function. Further, for the velocity field itself, the available choices are the single-relaxation time (SRT) and multi-relaxation time (MRT) models. Thus, for MHD-LBM, there are four possible permutations: SRT-BFF, SRT-EEF, MRT-BFF and MRT-EEF. Numerical implementation of the first three have already been presented in the literature. In this work, we, (i) develop the numerical implementation of MRT-EEF and (ii) perform an assessment of the four possible approaches. Our results indicate that the MRT-EEF is the most robust and accurate of the MHD-LBM computational schemes examined.