Say you hold out a laser pointer up against the moon like your short piece of string and get the laser pointer to point upwards. Assuming no atmosphere absorbency, if you could turn it on would its photons broadcast out into to space away from the earth or would they curve around on the celestial sphere and hit the sun on the horizon behind you?
Right. So you're going to have to define what you mean by "upwards". That might sound like a ridiculous request, but the whole point of this illusion is that it only
appears that the moon's terminator points upwards. You surely must be familiar with optical illusions, our brains can be easily fooled in certain situations. If you had a laser pointer on a day when this optical illusion occurs and you lined it up with the moon such that the laser shines perpendicular to the moon's terminator then if you could see it the laser line would intersect the sun. I know it doesn't look like it should but that is literally what is meant by an optical illusion.
Let's see if this helps. Here's a row of streetlights.
Because of perspective they get taller as they get nearer to you and so the line made by the light bulbs angles upwards from left to right, yes?
Assuming the row of lights continues if you turn your head to the right you would see the lights recede into the distance apparently getting smaller and lower as they go. So in that direction the line made by the light bulbs angles upwards from right to left. Agreed?
But what in reality does the line between all those lightbulbs look like? Assuming the lamps are all the same height and the road is straight it's a straight line, it's only perspective that makes it appear to be more of arc.
I think I realised what you've done when investigating this illusion. You simply stretched a bit of string out perpendicular to the terminator but you've only stretched it a short distance, inferred that the short length of string shoots off into space and you haven't stretched the string any further to determine where the line actually continues. Had you done so you'd have realised that it does point to the sun. You say that what you did is a better experiment but it's actually worse. It's worse because you are trying to infer where the line will continue rather than extending the string and finding out where it continues. We are pretty terrible at that sort of thing as demonstrated by the Poggendorff illusion. If you continue the black line through the occluding grey bar, which line is the correct continuance of it.
Most people perceive it as the blue line when in fact it is the red which you can easily demonstrate by drawing a contrasting line on top of the black one to see which line continues the straight line:
You are inferring the line shoots off into space because that's what it looks like. But that's the exact point of this optical illusion. The only way to find out where the line really goes is to stretch enough string to find out. Do that and you'll see that contrary to appearances the line does indeed hit the sun.