You sound like someone who can’t tell the different between a griffon, a griffin, and a hippogriff (note : the difference between the first two is in the legs, as griffons generally have lion limbs, while griffins have avian limbs… and a hippogriff has equine features, obviously)… or for that matter between an Each-uisge (large bodies of water) and a Kelpie (running water).

Clarification… the whole rooster top-half, snake base-half routine is definitely Cockatrice. The Basilisk is just a lizard full stop, regardless of system. Not sure where the D&D one got extra limbs from, mind you.
BUT the Basilisk is the one that can petrify with a stare (along with Gorgons and Catoblepas). The Cockatrice’s stare just causes death, decay, that sort of shenanigans.