"Carnelian" and "cornelian" are two different words for the same material. They are not two different spellings of the same word. Cornelian is British, and Brits will be the first to tell you their version is the correct one, regardless of Beck or anyone. (Based on my experience.)

Your gemological exposition is not exactly accurate. Chalcedony is a variety of quartz. Specifically, a microcrystalline quartz of a fibrous structure, giving a particular hazy (not clear and not opaque) diaphenaity. Agate is not necessarily a subclass of chalcedony, though it might be. Agate is a microcrystaline quartz that typically has a banded structure. Some of these bands are chalcedony. When chalcedonies and agates are largely red or reddish, they are called "carnelian" or "cornelian."