Well thank you to everyone. At long last I fully understand this problem.

The sky is not flat. When we look at the sky we're essentially looking out at the internal surface of a sphere (like when you're in a planetarium). And when you project a straight line onto the internal surface of a sphere it appears as an arc (unless you happen to be located in one of a special set of locations where the line appears straight, but slightly shorter than it really is). And so the straight line between the moon and the sun appears as an arc in the sky (unless you're near the equator) and the moon's terminator is orthogonal to this arc.

You can demonstrate this arcing of a straight line in the comfort of your own home. Go and stand halfway along a long internal wall in your room, facing the wall and about 30cm away. If you let your eyes follow the straight line of the top of the wall from far left to far right (you're allowed to move your head) your line of sight actually follows an arc through the air, because you have to tilt your head back as the line passes in front of you and then tilt it forwards again as you follow the line to the right. So it's a straight line that appears as an arc.

At least we've discovered that it's got nothing at all to do with large distances or relative distances or relative sizes. It's just cartesian vectors expressed spherically.

Thanks again,

clop