I use them interchangeably, which may not mean much, but I'm usually more rigorous than suppliers seem to be. "pygmy octo" or "dwarf octo" could be just about anything, although they're more likely to be small adults than baby medium or large octos. It helps a lot more in IDing if they'll tell you where they were caught, since bimacs and briareus are relatively easy to tell from the dwarf species in the same areas.
I also use the two terms interchanably and don't know if there is a formal difference. The most common pygmies in the aquarium trade are the Caribbean, O. mercatoris. These are nocturnal and do not learn to interact early in the evening (unlike some of the medium sized octopuses from the same area). Mine have learned to come out for feeding around 11:30 (two will come out earlier if the lights are turned off by 10:00 but not so with the other 3). They are typically very shy and not very human interactive but I believe tank size and human exposure time do have some impact on their willingness to be comfortable around people. One of my 5 is quite social, his tank mate less so but has learned to tolerate the food person and will remain in the open after lights out. The three in my larger tank try not to be seen for any longer than it takes to get their food.
The Joubini, as I understand it (I have not knowingly seen one in person) cannot be distinguished from the Mercatoris by the casual enthusiast unless it lays eggs. Mercs are a large egg species (therefore mine are mercatoris ) and the Joubini, a small egg species. Similarly the bimaculatas and bimaculoid are hard to distinguish but produce small or large eggs, defining the difference.