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.
There is no consistent pattern to the names of the lunar phases. There are (at least) two separate reasons for each of the names, but there doesn't appear to be a reason for which name is chosen. The choice of whether to describe them based on their instantaneous appearance or based on their phase (!) in the cycle does seem arbitrary. I'll continue to think of them as arbitrary until someone provides a description of the pattern for choosing those names, and/or a reason for calling a half-illuminated view a "quarter" while also calling a fully-illuminated view "full.
-- ghunchu'wI'