We perform molecular dynamics simulations of the phase separation processes of a binary quadrupolar fluid that shows a closed-loop phase diagram with two types of critical points. These simulations are aimed to clarify the difference in the behavior of spinodal decomposition around the two types of critical points. After abrupt temperature jumps from the initial stable one-phase states outside the closed-loop to the same unstable state inside it, we track the time evolution of the fluid on multiple length scales: the direct observation of the whole system, the structure factor for the binary fluid, and the local structures formed by the quadrupolar molecules. In the very early stage of both kinetic pathways, we observe either the rearrangement or destruction of self-assembled molecular aggregates within the same characteristic time scale, followed by the usual spinodal decomposition toward the same final equilibrium state. In conclusion, the phase separation processes near the two types of critical points are qualitatively the same.