Looking at the actual graphs, what is presented appears at first blush to be a bistable system prepared in its higher-energy stable state. (Or at least, prepared in a way that several of its constituent subsystems "fall into" that state.) Transitions to the lower-energy stable state require overcoming a barrier, and jumping over that barrier is of course faster at higher temperatures. At lower temperatures you essentially have a slow internal self-heating from the periodical jumps of subsystems through this forbidden region by thermal excitations, not unlike phosphorescence having to tunnel through an energetically forbidden region; this self-heating prevents the thing from cooling down fast enough.
Actual applicability to water is kind of harder to evaluate. I've long thought that you could indeed have an Mpemba effect caused by persistent induced convective currents in a fluid: so a hotter thing placed into a fridge might create a more dramatic internal flow as its outer boundary layer cools and changes in density, the convective current would certainly increase the slope of the hotter cooling system beyond the naive temperature scaling, and might then persist as a physical difference even after the two systems arrive at the same temperature, leading to a faster cooling speed of the convective system that "started out hotter". But I mean I have never really done experiments to confirm that sort of thing.