Is there any distinction between pressureless, collisionless and self-interacting dark matter or does being one imply the other?
Self interacting: the particles of DM interact between each others, they can bounce and scatter (they do not have to do it necessarily though);
Collisionless: the DM particles density is so low that they never interact; no collision between them happens;
Pressureless: the volume over which your ensemble of DM particles is distributed can be shrunk without doing any work;
Generally self interaction will cause collisions (violaing collisionless hypothesis), but there can be regimes (e.g. low density and temperature) where collisions never happen all the same. Pressureless-ness looks like a more model depending feature, but in general it will be different from the other two: the pressure will be measured by the amount of kinetic energy you have to give to the particles in the ensemble as they bounce on the boundary that you are moving (shrinking the volume); this will be non zero even if the particles never hit each others.
So they are three different characteristics (and conflicting in the case of self interacting and collisionless-ness).