For it to count as a kill, the finishing blow must be made through a regular or critical strike with the Chaosbringer.
WEAPON SKILL KILLS DO NOT COUNT. Must kill with a regular or critical strike as stated above.
As long as the finishing blow is dealt with the Chaosbringer, the method(s) used to damage the monster are irrelevant. For this reason, WAR/BLM casting Black Magic to weaken enemies, with Fields of ValorRefresh active, is a very effective strategy for low level Warriors.
During Harvest Festival events in real-life October, the Wraith Bats that spawn outside the starter nations have only 1 HP, making them the ideal target for this quest.
Weapon skills that defeat the monster, or abilities that do so (such as Jump), do not count towards the kill total.