Haha, well I’ve always thought of them as “modular medical/engineering tools,” based on Rick Sternbach’s original drawing (“modular medical instrument”) but that doesn’t quite seem techno-babble-y enough. The repainted screen used forked one was referred to as an “isolinear actuator” in a yourprops entry but I don’t know where that name came from as there’s no reference to it anywhere else.
The light-up one is called an “exoscalpel” in the episode “Ethics” so that sounds good to me.
I forgot that there’s a third version of the case - a slight variation on the three-hinge model which has molded in “feet”:
It is this molded-feet version that doesn’t like to open the full 180 degrees. Both three-hinge versions tend to develop chips in the plastic at the edge near the hinges, though, which is likely why that area was modified and thickened in the piano hinge verison. Chips are visible here, along with a photo of the thin plastic holding the hinges snapped into place: