Could you explain the difference between vengeance and ‘avenge’ (see Luke 18:3 KJV)?

This text is from a transcript of a talk by David Gooding, entitled ‘The Kingdom of God’ (2004).

Revenge is if you hit me on the nose, I get my revenge and hit you on the nose, possibly twice over for good measure!

If somebody pains you and you get your own back on him, that’s revenge. But when you take your case to the court and the judge gives sentence, he’s not executing revenge; he’s the impartial judge, but he’s seeing that justice is done: that’s avenge. The Greek has different words for it so it makes it clear: our language has its limitations that confuse people!