A pronounced difference in the mechanism of Li and Mg-ion insertions into the Chevrel phase (M x Mo6S8, 0<x<2 for Mg and 0<x<4 for Li) has been found using electrochemical impedance spectroscopy (EIS). Whereas the Li-ion insertion mechanism was qualitatively similar to that for a variety of transition metal oxides (charge-transfer semicircle+finite space Warburg), Mg-ion intercalation into the Chevrel electrode revealed two new features: a Gerischer-type impedance in the high-to-medium frequency domain and a low-frequency semicircle due to charge trapping. Evidence was provided in favor of a solid-state origin of the Gerischer impedance linked to a following chemical reaction (with respect to the Mg-ion-transfer) between the doubly charged cation and the S8 anion. Based on a transmission line representation, a more general model for charge trapping has been proposed, which reduces to a Gerischer impedance in the high frequency limit. Charge trapping (i.e., the solid-state ion association) was recognized to be the major reason for a reduced Mg-ion mobility in the Chevrel electrode compared to that of Li-ion. We also discuss a peculiar potential dependence of the resistance of ion-transfer into the Chevrel electrode in the framework of a lattice-gas model with elastic interactions and stresses in the host bulk.