It can be fuzzy and arbitrary at the same time.

It's certainly arbitrary that the Gulf of Mexico isn't considered a sea. (or various other "parts of the ocean" referred to as "gulfs" or "bays")

"The main difference between a sea and a lake is that a sea is a part of the ocean."

Ok, so you have decided that the Caspian Sea isn't actually a sea? Well I guess all bets are off if you can do that.... :)