We derived one-fluid equations based on a relativistic two-fluid approximation of e$^\pm$ pair plasma and electron-ion plasma to reveal the specific relativistic nature of their behavior. Assuming simple condition on the relativistic one-fluid equations, we propose generalized relativistic magnetohydrodynamic (RMHD) equations which satisfy causality. We show the linear analyses of these equations regarding various plasma waves to show the validity of the generalized RMHD equations derived here and to reveal the distinct properties of the pair plasma and electron-ion plasma. The distinct properties relate to (i) the inertia effect of electric charge, (ii) the momentum of electric current, (iii) the relativistic Hall effect, (iv) the thermal electromotive force, and (v) the thermalized energy exchange between the two fluids. Using the generalized RMHD equations, we also clarify the condition that we can use standard RMHD equations and that we need the distinct RMHD equations of pair and electron-ion plasmas. The standard RMHD is available only when the relative velocity of the two fluids is nonrelativistic, a difference of the enthalpy densities of the two fluids is much smaller than the total enthalpy density, and the above distinct properties of the pair/electron-ion plasma are negligible. We discuss a general relativistic version of the equations applicable to the pair and electron-ion plasmas in black hole magnetospheres. We find the effective resistivity due to shear of frame ragging around a rotating black hole.