As OddBall points out, Stick Saver and Rim Shot hoops are the same thing. Slingerland officially referred to them as "rim shot counter hoops", but never any mention of "stick saver". The "stick saver" moniker comes into play with the fact that, while being a triple-flanged hoop, the final flange is turned inward towards the head, rather than outward as used by the rest of the industry. This inward flange really did "save" sticks!
Stick Chopper hoops were a double-flanged hoop. And, yes, they sure could chop up sticks!
I believe that the COB Rimshot counter hoops were used to about 1969, when they were replaced by COS hoops. These first COS hoops were obviously made on the same equipment as the COB hoops, as they have the same height and profile as COB hoops. Over the years, the COS hoops began to take on different profiles!
So, there you go. More "hoop" info than you ever wanted to know!
-Mark