I noticed that there were a lot of posts about this sonata, and I tried reading as many as I could. My comment is that I have been playing the 1st movement of this piece for a long time, (over 50 years), but have come to learn that Beethoven never called it the "Moonlight", that happened about 4 years after he died. My thought was to put that impression out of my mind, and just listen to the music in light of Beethoven's actual title "Sonata quasi una Fantasia", and come to my own conclusions. In the last few years I have decided that the piece sounds more like a rainstorm than moonlight on a lake. With that in mind the other movements fit a little better, with the second being some children playing in the wet grass, and the third a really violent storm. I would be curious to know if anyone else has come to this conclusion, and I'm sorry if I have duplicated a previous post or thread, but I haven't had a lot of time to search the forum.
I agree completely with what your saying if memory serves me correctly it was a poet somewhat slightly later who coined it as the moonlight sonata as it inspired a poem he wrote (I presume the poem was called moonlight).However a fantasia or fantasie is a more fitting title for it as it really has no resemblance to moonlight (first movement aside) where if you compare it to much later works like Debussy and Faure who actually title pieces moonlight I'm sure you've heard of one of them....
I would suggest that Claudio Arrau, has done a great service for pianists by demonstrating that pieces do not always need to be played very fast, and sometimes played slowly allows for more expression.