I always assumed that it just meant that any interactions they had with non-dark matter were weak if they existed at all. There is also the issue that WIMPs are not expected to form large clumps, but rather to exist in diffuse halos around galaxies. I suppose that might imply that their interactions with each other were also weak. I didn't think it meant anything to do with the weak force, although the Wikipedia page says that interacting with the weak force might be a possibility, just that any theoretical weaker interaction also counts. The only issue is that it would still have to be much weaker than the weak interaction of left-chiral neutrinos and right-chiral antineutrinos, since we've detected loads of those and those are not considered to be a dark matter candidate anymore. The only reason I can think of why they would have to interact with the weak force, or rather the electroweak force, is that I believe that's the type of interaction that the Higgs field is believed to provide elementary particles with mass. I don't think it's clear that that's the only way particles could get mass, though, so even that doesn't seem like very appropriate except for a more specific class of theories.
In short, I don't know if someone uses it to refer to something involving weak interaction, but there is probably also a broader usage of the term too, even if that is a usage, since the general astrophysical/cosmological context that I've heard about it in made a three-way contrast between WIMPs, MACHOs, and alternative gravity and fifth force theories (I guess that's technically 4-ways) as the main classes of possible explanations, with alternate gravity theories being mostly killed by the detection of highly dark-matter deficient galaxies and MACHOs being less popular because star-occultation/brightening studies (which, of course, assume assume either fairly strong interaction with light or, more likely, just gravitational lensing, which would even be expected of particularly large and dense clumps of dark matter that didn't interact with photons). The space that WIMPs occupy in this broad view is basically that of the idea of dark matter being any kind of exotic matter, presumably from outside of the standard model, that fits the astrophysical description of dark matter by it's very nature, and so does not need to be clumped up into tiny volumes to explain why we can't see it. I don't know if that's an over-generalization, though: I'm not an expert.
(I was going to post this as a comment, but I wanted to write more than would fit in one, and it is basically an answer.)
