In the end, we (Joel Kamnitzer, his student Bruce Fontaine, and I) agreed on the term "diskoid". In the abstract, the word "cactus" seemed too clever by half. When I actually wrote it into the paper, it was awkward and it didn't emphasize the main property of interest, that our shapes are meant as a mild generalization of disks. Moreover, there have been papers that used the term "cactus" in similar but slightly different ways, and the mathematical etymology of this term is already somewhat confused. Actually, I don't have a good explanation of why the term that we picked seems good (or least bad), but somehow that is the feeling.