Let's get right back to the several specific and sensible questions asked by the originator of this thread:
1. In vocal music, it is accepted that there is a story. Movements are trained, expected and accepted in singers to help convey the story of the music. Even choreographed, including specifics about eye focus and so on. Vocalists are expected to act. Why is this un-acceptable in piano performance ?
It is not so much "unacceptable" as unnecsessary, for two reasons. Firstly, the singer's sounds come from the singer him/herself - in other words, his/her body is itself the instrument that produces the sound - whereas the instrumentalist uses parts of his/her body to operate a piece of independent machinery (if I may be forgiven for describing the greatest Cremonese stringed instruments or Viennese pianos as "pieces of machinery"). Secondly, the singer usually sings words, whose meaning, by definition, is independent of musical meaning and sufficient of itself as verbal expression; when words are set to music, the end product is usually a composite expression involving both words and music in which the whole is greater than the sum of its parts, whereas instrumental music, even when programmatic (which is often not the case anyway), conveys its message by means of musical sounds alone.
2. Just because there are not words in a solo piece of piano music, does it really mean there is not a story that needs to be acted out ? And why would we as pianists be any less responsible for conveying that story than a vocalist ?
See 1. above. Of course it does not mean that there is no "story" as such; even if the piece is non-programmatic, it may tell its own "story", but it does so without using words and so cannot be accounted for in words - accordingly, the music does not of itself call for the performer to "act", in terms of adding gesture to assist in conveying what it expresses.
3. Is our body any less a part of our instrument and even character of a piece as a pianist vs that of a vocalist ? If so, why ? If not, why not use it to help us express the music and the story of the music, clearly ?
Again, see 1. above. When performers are not making sounds with their bodies - i.e. singing - and are presenting an expression by means of music alone rather than in combination with words, then it seems fair to ascribe far less expressive relevance to instrumentalists' body movments as to those of singers; the instrumentalist needs to make all the moves necessary to produce the sounds from his/her instrument, but that's the extent of necessary movement for such a player.
4. Is singing supposed to be more visual than piano-ing ?
Whether or not it is "supposed" to be so, the above hopefully demonstrates that it would to some extent appear to be so. The reason I add the phrase "to some extent" is that I am mindful of the contributor to this thread who put forward a very valid argument about recordings; clearly, when one listens to an audio-only recording of music, whether sung or played, any visual element is lost to the listener in any case, so any performer - singer or instrumentalist - has, since the dawn of recorded music, had to bear in mind that his/her performance must be sufficiently expressive by dint of musical sounds alone.
Another factor here might be that of perceived value judgements. It is not inconceivable (well, almost not!) that a physically demonstrative pianist might give as powerful a performance of, say, Rakhmaninov's Fourth Piano Concerto as the famous one by Michelangeli; if two such performances were of equal validity and greatness as expressions of the composer's intentions, one might be tempted to ask oneself why the demonstrative performer needed to make many of his/her extraneous movements if Michelangeli (or the composer himself, for that matter) could achieve equally fine results by means of the greatest possible economy of body movement.
Best,
Alistair