Combining robustness and high accuracy is one of the primary challenges in the magnetohydrodynamics (MHD) field of numerical methods. This paper investigates two critical physical constraints: wave order and positivity-preserving (PP) properties of the high-resolution HLLD Riemann solver, which ensures the positivity of density, pressure, and internal energy. This method’s distinctiveness lies in its ability to ensure that the wave characteristic speeds of the HLLD Riemann solver are strictly ordered. A provably PP HLLD Riemann solver based on the Lagrangian setting is established, which can be viewed as an extension of the PP Lagrangian method in hydrodynamics but with more and stronger constraint condition. In addition, the above two properties are ensured on moving grid method by employing the Lagrange-to-Euler transform. Meanwhile, a novel multi-moment constrained finite volume method is introduced to acquire third order accuracy, and practical limiters are applied to avoid numerical oscillations. Selected numerical benchmarks demonstrate the robustness and accuracy of our methods.