I'll throw a 97% save percentage out the window, b/c frankly any pitcher saving 97% of their games is not going to be getting by allowing a runner or two every other appearance.
Coco was pretty solid during his Reds tenure. he saved something in the neighborhood of 88% of his chances for the Reds. he was at 83% then 86% in his last 2 seasons. It would drive many fans nuts the amount of times he'd walk a guy or give up a hit, or even give up a run when coming into the game w/ a 2 or 3 run lead before closing it out. I can't help but imagine that the starting pitchers felt the same uneasiness, carried the thought with them that they needed to last as long as possible to ensure that they'll get the win.
Meanwhile, Aroldis has saved 88% of his chances this season. You don't think it sits in the back of Cueto's mind when he took the mound in the 8th against the cubbies w/ a 3 run lead the other night that all he needed to do was get the next 3 outs. He could use everything he had left, b/c he knew the 9th was safe.
Perhaps I'm giving that mentality too much credit, but I would think it would have to make a worlds of difference knowing you've got an absolutely guy w/ absolutely filthy stuff like Chapman vs a guy you felt was "just getting by" like Coco often appeared to do.