I agree with most of your comments, but Ringo doesn’t qualify because he’s British. But mentioning Ringo - who covered “Matchbox” and “Honey Don’t” - calls to mind another very cool guy, Carl Perkins. As for Jim Morrison, I disagree; he was too out-of-control to be cool, and his death bordered on suicide, which - by both of our criteria - would disqualify him. Of The Doors, I always thought that Ray Manzarek (RIP) was Mr. Cool, followed by Robby Krieger.
Did you ever see the video of The Doors, probably from a TV show, where Krieger has a black eye?
Regarding Carl Perkins; He was a gentlemen whom I had the pleasure meeting and also watching him, Joe Ely and Delbert McClinton at an after hours jam session in Dallas (Nick's on Greenville). Sun was coming up on one heck of a night as everyone left.