They are all one species, betta splendens. They only have tail types, no breeds whatsoever. Like for fancy mice, all the same but different variety.
VT x CT = VT with web reductions. The first few generations will have uneven but if you line breed the reductions will even up through time.
CT x HM = HM with web reductions like above. ^ Through time, it will be a HM finnage with spiky reductions of a CT. Also known as "Halfsun" or a more preferred term, Combtail.
HM x VT = I'm not really sure, but I think it's a HM with a rounded caudal.