As used in the OP's recommended video, to "run" a shotgun means all aspects of shotgun operation --including the choices of what gun to buy, how to modify it or what accessories to add to it, what ammo to feed it, learning the basic manual of arms, such as loading and unloading, using the gun's safety features, holding the gun in the correct stance and grip, and firing it quickly with practical accuracy.
As used in this thread AFTER page 1, we are referring to "running" a shotgun as to what grip affords the best CQB shooting-- the fastest, most stable, most accurate, and without smashing your own thumb into your nose under recoil.
As used in this thread AFTER page 1, we are referring to "running" a shotgun as to what grip affords the best CQB shooting-- the fastest, most stable, most accurate, and without smashing your own thumb into your nose under recoil.