I can't speak for an Uberti and how they function. I have never seen one that was anywhere nearly as reliable as a Colt or USFA. Whenever I have been at cowboy shoots, all of those copies have performed poorly, although I haven't been to one where a USFA was being used.
I would also doubt that your '97 is slam firing. If the bolt lock is working, the hammer shouldn't follow the bolt into battery without the bolt closing first. You would need to slowly pump the gun with the trigger pulled and watch how far the bolt goes before the hammer falls. If the hammer follows the bolt all the way to fired position, there is something wrong with the shotgun.
As far as a term for the correct mechanical process of the gun when the trigger is held down and the pump being worked, I am not aware of it.