Your definition of revenge is my definition of justice in this case, because society at large is OK with it and not just the victim's relatives.
"Justice—as logically, legally, and ethically defined—isn’t really about “getting even” or experiencing a spiteful joy in retaliation. Instead, it’s about righting a wrong that most members of society (as opposed to simply the alleged victim) would agree is morally culpable. And the presumably unbiased (i.e., unemotional) moral rightness of such justice is based on cultural or community standards of fairness and equity. Whereas revenge has a certain selfish quality to it"