If it was simply the wrong size, it would be like putting 32 oz in a 1 oz cup, or 1 oz in a 32 oz cup. To me that would be the wrong size. I think "half full" and "half empty" are both correct. Unless you know if the cup was being filled or emptied.
If I was on the side of a mountain, in the middle between the top and the bottom. Would I be "half way" up the mountain or "halfway" down the mountain? If you asked me, it would depend where I started. If I got there from the top of the mountain, then I would be "half way" down the mountain. If I started from the bottom, I would be "half way" up the mountain. If I asked you, it would depend on where you were, or if you knew my direction of travel. That would determine the correct answer.
So, if you knew if the cup was being filled or emptied, that would determine if it was "half full" or "half empty". If you didn't know the direction, both answers are right, rather than neither.
To me "near" and "nearly" have two different meaninings. A "near miss" would be a miss, but was near or close. "Nearly miss" would be to almost miss.
Right now I am nearly confused and half crazy.