I can now tell you all how this light works. There is a chamber a little more than twice as long as the length of the opening aperture of the skylight below. At the far end of this chamber there is an LED lamp with a collimating lens in front of it, just like a projector. This beam shines horizontally through the chamber. The chamber is filled with water with a cloudy suspension of nanoparticles. These act to cause Rayleigh scattering on the beam of light. Then a mirror over the skylight directs the beam downwards. The angle of the mirror can be electronically adjusted so as to adjust the angle of the beam of light (giving the appearance of the sun slowly moving across the sky).
The horizontal length of the chamber, as well as the collimating lens, make the beam of light very collimated into relatively linear beam, making it appear that the light in the skylight is much further away than it actually is. This helps create the illusion of depth because that is how sunlight behaves. The process of Rayleigh scattering, which mimics the natural effect of light passing through the atmosphere, also has the effect of shifting the spectral wavelength distribution from the LED source. This also has the effect of helping to make the light appear more natural. The chamber of course is filled with blue light from the scattering as the beam passes through it. This light constitutes the background bluish sky light.