Dynamical behavior of vortices plays central roles in the quantum phenomena of two-dimensional (2D) superconductors. The quantum metallic state, for example, showing an anomalous temperature-independent resistive state down to low temperatures, has been a topical subject in recently developed 2D crystalline superconductors, whose microscopic origin is still under debate. Here, we unveil an interesting aspect of the vortex dynamics in a noncentrosymmetric 2D crystalline superconductor of gated MoS2 through the nonreciprocal transport measurement. The second-harmonic resistance R2ω at low temperature with high current indicates the classical vortex flow accompanying the ratchet motion. Furthermore, we found that R2ω is substantially suppressed in the quantum metallic state with the low current region, allowing identification of the quantum and classical ratchet motions of vortices by the magnitude of the second-harmonic resistance. Combining R2ω with the linear resistance Rω allowed us to establish a comprehensive vortex phase diagram for crystalline 2D superconductors with minimal disorder.