you are definitely not getting the support you think from those definitions. vigilantism fairly clearly has nothing to do with a commensurate response.
So 'avenge' doesn't ring any bells with you? True justice is not about vengeance. In America, we have specific clauses forbidding 'cruel and unusual punishment'.
I'm sorry it seemed that way to you. I try to be concise in writing. If it has an emotional tone, its not because I meant to put it there.
The adjective form of vigilante spells out violent and summary action. Summary action is often incorrect action which would not be commensurate with the (non-existent in that case) crime. Violent action is not the usual response of the justice system, so again it would be a response not commensurate with the (real this time) crime.
That definition seems to support my original description pretty well!