HauntHaunt is a triggered ability. “Haunt” on a permanent means “When this permanent is put into a graveyard from the battlefield, exile it haunting target creature.” “Haunt” on an instant or sorcery spell means “When this spell is put into a graveyard during its resolution, exile it haunting target creature.”
Cards that are in the exile zone as the result of a haunt ability “haunt” the creature targeted by that ability. The phrase “creature it haunts” refers to the object targeted by the haunt ability, regardless of whether or not that object is still a creature.
Triggered abilities of cards with haunt that refer to the haunted creature can trigger in the exile zone. (CR 702.55) (When this creature
diesIs put into a graveyard from the battlefield. (CR 700.4),
exileTo exile an object, move it to the exile zone from wherever it is. See rule 406, “Exile.” (CR 701.13) it haunting target creature.)
When this creature enters or the creature it haunts
diesIs put into a graveyard from the battlefield. (CR 700.4),
destroyThe only ways a permanent can be destroyed are as a result of an effect that uses the word “destroy” or as a result of the state-based actions that check for lethal damage (see rule 704.5g) or damage from a source with deathtouch (see rule 704.5h). If a permanent is put into its owner’s graveyard for any other reason, it hasn’t been “destroyed.”
A regeneration effect replaces a destruction event. See rule 701.19, “Regenerate.” (CR 701.8) target creature that was dealt damage this turn.