Starlink (and other satellites) categorically do not have exterior lights or illumination, that would be a waste of power for no particular benefit.
The reason that we can sometimes see satellites or other spacecraft at night is because the spacecraft is still exposed to daylight. Here is an image describing this phenomena:
(image credit: Gary Meader; from the book "Night Sky With The Naked Eye" by Bob King)
Satellites and other orbiting bodies (ie the Moon) appear so bright because they're reflecting sunlight. Even though they aren't covered in mirrors, the sun is just so bright that white paint or the lunar surface is enough to let us see even tiny satellites from hundreds of kilometers away.
Occasionally, a satellite will "Flare" which is when the angle of the sun on its glass solar panels reflects directly to the observer. This reflection off the solar panels rather than simply the body of the satellite can appear much brighter temporarily.