Not all Cockatrice count for "Vanquish Cockatrices (UC)"

[Lizardman]

Dune Cockatrice in Abyssea-Altepa do not count towards the Records of Eminence quest "Vanquish Cockatrices (UC)" under "Unity", "Unity (Ayame)".

[Zarchery]

That's how those objectives work. Abyssea doesn't count for any of the Vanquish objectives.

[Shirai]

Abyssea is excluded from pretty much all "kill" records of eminence quests, under which the Unity quests also fall.