These items are updated!
Regarding Eagle Eye and White Castle:
We've been going with "the Eagle Eye" because it's a one-of-a-kind, kind of famous monster. It's a griffon, but it's title that people call it is "the Eagle Eye," kind of like how "the Blue Light" the title of the demon king (and Elyas' sister).
As for White Castle, despite the name, it's not just referring to the castle itself, but also to the surrounding area and probably even the town that used to be near the castle back when people lived in Huzak.