First, Braid, where your character can die in the usual way for a 2D platformer, killed by monsters or falling in holes. But the game lets you manipulate time, so the answer to death is just to rewind a few seconds while the music plays backwards and you come back to life.
Second, the original Monkey Island. Unlike the Sierra click puzzlers of the time (King's Quest etc) you couldn't die. But if you stood at the edge of the cliff, you fell off and got a Sierra-style Game Over screen... for a few seconds. Then you bounced back up on a 'rubber tree'.
Finally, there's the original ZX Spectrum isometric platformer version of Batman. In that game, if you died in a particularly frustrating way or at the same puzzle repeatedly, the game would sometimes give you a "Dog's life", an extra life as a consolation prize.
Interesting article.