If we give so that everybody can see, there is a tempory improvement on how other people perceive you and in fact you have done good to others.

People have short memories and sometimes brief lives and will forget your good deeds. Therefore for motives of just doing good deeds, we don't need to be identified by anybody but by God who will take note.

God says Vengeance is mine yes true BUT we also have a right of self defence and preservation of own lives and that of innocent people.

What that is saying is dont revenge the evil God will take care of that but in a moment of decision making where there there is a real and impending danger of harm or life defend yourself