Calling it “Full” is based on its appearance. There’s nothing in its status within the cycle that has a reason to be called “Full”. 

Calling it Quarter is based on its status within the cycle. It has nothing to do with its appearance. 

Calling it “New” is arbitrary, but you could think of it as the zero point of the cycle, but if it was, then the Full Moon would logically be called the Half Moon, and the fully waned Gibbous would be the Three Quarter Moon.

It’s arbitrary. It may be based on a “deeper understanding of the Moon than its appearance, but it’s still abitrary.

You keep using that word. I do not think it means what you think it means.

Saying something is arbitrary means there is no reason or pattern to it; it's just based on whim or randomness. But these names are not without reason or pattern and aren't based on whim or randomness. They were named these things for reasons, and the terms have a long linguistic development.

What you apparently mean by "arbitrary" is that the names don't conform to what you you happen to think would be the most logical system for naming the phases of the moon. They're not arbitrary in the same way that the silent gh in English knight isn't arbitrary. It's there for a reason, whether or not you know or like the reason.

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SuStel
http://trimboli.name