All 4 Uses

vengeance
in
The American, by Henry James
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You are a man of the world with a vengeance!†
Chpt 7 *

with a vengeance = with intensity
Newman strolled lightly beside her; his head was thrown back and he was gazing at all the stars; he seemed to himself to be riding his vengeance along the Milky Way.†
Chpt 22 *

vengeance = the act of taking revenge
He went home, and feeling rather tired—nursing a vengeance was, it must be confessed, a rather fatiguing process; it took a good deal out of one—flung himself into one of his brocaded fauteuils, stretched his legs, thrust his hands into his pockets, and, while he watched the reflected sunset fading from the ornate house-tops on the opposite side of the Boulevard, began mentally to compose a cool epistle to Madame de Bellegarde.†
Chpt 23
But they were frightened," Newman added, "and I have had all the vengeance I want."†
Chpt 26

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)