All 3 Uses

vengeance
in
The Jungle, by Sinclair
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Jurgis was called upon to do the beating, and as it hurt his foot he did it with a vengeance; but it did not tend to add to the sweetness of his temper.†
Chpt 12 *

with a vengeance = with intensity
And every hour his soul grew blacker, every hour he dreamed new dreams of vengeance, of defiance, of raging, frenzied hate.†
Chpt 16 *

vengeance = the act of taking revenge
It will be a movement beginning in the far-off past, a thing obscure and unhonored, a thing easy to ridicule, easy to despise; a thing unlovely, wearing the aspect of vengeance and hate—but to you, the working-man, the wage-slave, calling with a voice insistent, imperious—with a voice that you cannot escape, wherever upon the earth you may be!†
Chpt 28

Definitions:

(1) (vengeance as in: vengeance is mine)
the act of taking revenge

(Revenge means to harm someone to get them back for something harmful that they have done.)
(2) (with a vengeance as in: with a vengeance)
with intensity
(3) (meaning too rare to warrant focus)