We have Moment and Cantilever now.
Stiffener - Plate Steel welded between flanges
Braced Connection - structure between top flange and bottom flange of another beam
Drag - Beam that connects into another building frame and additional members extend bracing into the adjacent structure.
I suspect that one reason these other options don't exist is that external analysis tools either ignore these connect subtleties or deal with them differently enough that they've held off providing them. This is likely a "documentation" versus "engineering" situation...just a guess on my part.