Sure you did. We both know we are talking about two different things. The person put to death never kills again. He also doesn't take up needed space and resources. They also earned their punishment. Yes, I said punishment. If you kill without a just cause you should recieve just (fair or equal) punishment.
You are talking about what deters the nest person. Neither does because a person set on killing will do it no matter what. If the ultimate punishment doesn't prevent an action, nothing will. Anyone willing to take an innocent life is not concerned with justice.