And that's a big part of my question... given that in the abstract water can store any amount of heat (between absolute 0 and the point at which we can no longer call it "water" due to being a plasma of some form), what exactly does it mean for this material to be able to store 4 times as much?

I'm sure there's an answer, because I'm sure the journalist got that number from somewhere, but I lack the connections to know where to begin finding this information.