I'll hazard an attempt to articulate what Tom has been attempting to say is the explanation.
It's not a "literal spotlight." It's not the sun that is shaded from illuminating in certain directions. It's the atmo- (air) through which the sun is shining that is doing the shading.
When the sunlight has to penetrate less air mass, that is when we see it "projected upon the atmoplane." But at longer distances, when the angle of the sun is shallower, the air eventually becomes opaque to the sun and hence the sunlight won't penetrate it and for those in darkness, there is no "projection upon the atmoplane" due to that opaqueness.
This is not the cause of sunset. That's either a perspective effect or light bending upward. Even after sunset (or before sunrise) there is twilight because the sun's light is beginning to penetrate the air as the angle becomes shallower and thus less air to pass through.
If I have that wrong, I apologize; I can't claim to be able to fully apprehend it since I give it so little credence. It's more like a meteorological explanation than an atmoplanar one, and day/night would vary due to weather more so than characteristic atmo- opacity.
There's also the issue of the magnification effect claimed for which the atmolayer, with increasing density, magnifies the sun such that it maintains its apparent angular size despite receding into the distance. It's a remarkable confluence of features: atmolayer shading, magnification and perspective (or EAT) that causes the sun to appear to rise and set while all the while actually being some distance over a flat earth.
And this still doesn't explain the lateral displacement of the projected sun from the actual sun, but that's an issue pending explanation on another topic. Still, I hope I've gotten the "spotlight" "projection on the atmoplane" thing right.