Not sure the basewad makes a difference to the wad, they don't touch for any load I've ever done.
The 'insert' referenced would be the basewad.
Reifenhauser was the father of the plastic shotshell. He invented a way of work hardening and extruding a plastic tube that was both thin enough and strong enough to make a shotgun shell.
Those shells are still with us today as the 'Reifenahuser tube', are straight walled, and require a base wad of some type. The design of the shape of the combustion area and the internal volume varies considerably from brand to brand.
The 'compression formed' shell was devised by Winchester in the early 60's as another way to skin that cat and is made in a die. The shell is one piece with the exception of the head. The taper strengthens the base and improves the thermal efficiency of the shell. Remington STS and Gun Club are still one piece tapered shells. The AAHS is a hybrid of sorts and gets some of it's taper from the basewad.
All shells gas seal best and work the best with wads designed to match their internal dimensions. Other wads can be used and in fact many of the 'lower pressure' loads you see published rely on poor fitting wads. Powder migration can be a problem in such cases especially with fine grained powders such as ball powders. Flake powders mitigate this to some extent. It's still a dubious practice in my book.
