Snare ring has it's place. Live, I like some snare ring. I'm up against a lot of guitar wattage when playing live, and a full sounding snare definitely helps to cut through. I use no dampening when playing live - or maybe a half-piece of moongel on the snare up against the rim just to kill the highest overtones.
Recording is a different story. I want a dry, meaty crack with no ring whatsoever when recording, so I'll use a full moongel usually on top of the head logo - which is about an inch in from the rim.