As you admit, the experiment is using a perspective effect to get the illuminated portion of the ball to point upwards.
It's not an admission as such, more an explanation of why perspective is key in this experiment.
I don't really know what you mean by "perspective effect", you keep using that term as if it means something magic.
Perspective is important here for the reasons I've outlined. I've added a couple of additional "footballs" to my diagram:
You're looking from point A:
If you hold the ball to the left and the light source is to the right then you can only see the illuminated side of the ball.
If you hold it to the right then you can only see the dark side of the ball.
It's only if you hold it so it lines up with the moon that the line of sight is the same, the perspective is the same and the "phase" you observe on the ball is the same. Perspective is the reason we see different phases of the moon. In reality it's always half lit, half dark. But because of its changing position as it orbits us that changes our perspective so we see the different phases.
And one key thing to note here is of course the scale in my diagram is hopelessly wrong. In reality the moon is very distant - that's why it doesn't matter where you are on earth, you see the same phase. In the context of the moon being 250,000 miles away, a few thousand miles here and there doesn't significantly change your perspective (although the moon's orientation will be different in the southern hemisphere because you are "upside down" with respect to the northern hemisphere).
But the ball is very close to you so it's very easy to move it, or move your position relative to it so you observe different phases.
This doesn't demonstrate that the Moon is pointing upwards to a perspective effect. We are using a perspective effect to get the ball to point upwards, when it could also be that the Moon is pointing upwards and we are just matching it.
Right. So two experiments have been outlined and they demonstrate different things.
IF the sun is illuminating the moon AND the sun is distant so the light rays from it are in effect parallel AND light travels in straight lines
THEN if you hold a ball up so it lines up with the moon you should observe the same phase on the ball as you do the moon.
Because both the moon and the ball are being illuminated by the same light source, the rays are going straight and parallel, they're hitting the moon and the ball at the same angle and you're looking at both from the same perspective.
That's what my original diagram was trying to explain. Hopefully the new one with the extra balls shows more clearly why perspective is important.
The experiment is unable to distinguish whether the Moon is actually pointing upwards or not.
Correct. But that isn't what the experiment is intended to do. It's intended to add credence to the idea that the sun is distant and is illuminating the moon and the ball with parallel rays. I'll leave you to consider whether EA would make the same prediction.
There is another experiment we can do to determine whether the moon (more accurately the line perpendicular to the terminator on the moon) is pointing upwards. When you see the illusion the line perpendicular to the terminator looks like it shoots off into the sky. But as I've shown with the optical illusion above, we aren't very good at judging this sort of thing. I previously showed you this image:
It looks like the black line and the blue line join up when in fact it's the red line which is a continuation of the black line:
Humans pretty much suck at judging things in certain situations.
The point of the string experiment is that the string forms a straight line.
If the sun is illuminating the moon and light travels in straight lines then the line perpendicular to the terminator should intersect the sun. The string experiment shows that, contrary to what it appears to the naked eye, the string does indeed intersect the sun. Just like you can draw a line on the above picture to show that it is the black and red lines which form a straight line, not the blue:
And yes, the string experiment can't distinguish between whether the light is going in a straight line or whether it's arcing around a dome like in your water bottle diagram. But that isn't how light behaves either in the RE model where light travels in straight lines or in the FE one where light bends upwards. In your diagram the sun's light would have to be bending up and over a hemisphere which is the opposite of what EA claims.