Both Uses

vengeance
in
Treasure Island
(Auto-generated)

Soon after, with a drink all round, we lay down to sleep, and the outside of Silver's vengeance was to put George Merry up for sentinel and threaten him with death if he should prove unfaithful.†
Chpt 29 *

vengeance = the act of taking revenge
"Drink and the devil had done for the rest," with a vengeance, although, to be sure, we were not quite in so bad a case as that other ship they sang about: With one man of her crew alive, What put to sea with seventy-five.†
Chpt 34 *

with a vengeance = with intensity

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)