Alright, I think I am caught up. I wouldn't mind giving it a go if/when I am thinking about it and conditions allow for it. However, regardless of outcome, why would this be evidence for either side? Seems to be explainable in either model.
Junker, it is an invalid experiment. You are replicating the direction, but not the angle of the moon. With the string you may end up making a path that connects the two bodies, but is not true to the angle of the moon:
That is your misunderstanding in a nutshell. When the string or stick exactly bisects the sun and the moon, it does form a perpendicular to the line between dark and light on the moon, which you would have discovered if you were not so inexplicably resistant to actually looking for yourself. Maintaining that perpendicular relationship to the line on the moon, and keeping the line of the stick/string at the correct height so that it continues to bisect both the sun and the moon only allows movement in one additional plane which is to move one end of the stick closer or further away from your eyes (from above this would be rotating the stick like a pinwheel). But when you do that while maintaining the perpendicular to the line of light/dark on the moon, the stick or string still bisects the sun.
Changing this third dimensional position of the stick or string does not change the perpendicular relationship of the stick to the line on the moon, nor does it cause the line of the string or stick to stop bisecting the discs of the sun and the moon. Note, if you rotate the stick or string far enough away from your eye, it begins to appear shortened as any stick does if you turn it while keeping it at a preset angle due to perspective, but the actual line formed by the stick/string continues to bisect the sun and moon if it is extended beyond the ends of the stick/string which now appear shortened due to perspective. Of course if you keep rotating a stick like this, it will eventually form a point instead of a line as you will be looking at just one end of the stick, but prior to that point, the apparent directions extending beyond the ends of the stick will still bisect the sun and the moon and also maintain the proper angle to the light/dark boundary on the moon.
Please Junker, do the experiment and report back so that we can move on....although if we are going to wait for Tom Bishop to accept defeat, that is probably a pipe dream. Conditions will be ideal again around May 2nd when the moon will be at the first quarter phase, for easy viewing of the sun and partially shaded moon in the late afternoon and just before sunset.
PS: Tom, The problem with your drawing is that you are only showing two dimensions, so when the stick figure bends over, the perpendicular relationship of the stick to the light boundary on the moon is lost. This does not occur when actually doing the experiment, even when I turn the stick in the third dimension.