Karl Knechtel: I think a distinction needs to be made between moves that resolve a ko, and the more general class of local moves that can be played when you've "won" the ko. In particular, some 2-step kos are such because you must "win" twice to win a capturing race; others are such because you desire (but don't really need to) to fight on, risking the ko again to increase its value.
2 through 5 fight the ko anew after Black had "won" it. White concludes she has no more non-local threats, and extends. Black cuts feeling that he has even further ko threats. Now Black will be able to resolve ko by capture next time, due to the double atari, when he could only resolve by connection originally.