The bottom line is, if Beethoven wanted to give it a name, he would have, as he did "Les Adieux."
He gave titles to several of his works, so if he wanted to call op27#2 "moonlight," he would have.
There are lots of pieces with titles given to them that the composer never intended - "Winter Wind" etude, for example. I sometimes use those titles for the purpose if identification, but would never put them on a printed program, or other more formal identification.
I know I'm splitting hairs, but once you call a piece "Winter Wind," "Moonlight," "Tempest," etc. you are limiting the possiblilities of its interpretation. In any given piece, movement, section, phrase, motif, note, there are several emotions and images you can find. The last movement of op27#2 can sound heroic one day, tragic another day, playful another day. It depends on my mood. Why limit those possibilities with a label?
Don't mean to sound like a purist. Just my 2 cents.