BackWorking out the exact circumstances which give rise to illumination of the eclipsed Moon is somewhat tricky, as the amount of refraction is hard to estimate. We know that for an observer on Earth looking at a rise or set the refraction is about 34 arc-min. For an observer on the Moon however the value will potentially be greater, as the light gets refracted once when travelling from the Sun to the edge of the Earth, and again when travelling onward to the Moon - imagine the first diagram on the previous page with a mirror-image of the situation on the right-hand side.
Simple geometry shows that a beam of light just tangent to the edge of the Earth (as seen from the Moon) would have to be refracted by (on average) about 41 arc-min to strike the centre of the Moon at maximum eclipse. Although greater than the 34 arc-min terrestrial figure, it is clearly not so much greater that it could not be produced by the extra path-length referred to above. The Moon will therefore always receive some refracted light, throughout the eclipse. However, that does not mean that an observer on the Moon would always be able to see Sun's disc, as the overall geometry is different - a lunar eclipse on Earth is a solar eclipse when seen from the Moon. What would be seen is a "halo" around the Earth, which would be about four times as large as the Sun in the sky rather than, as seen from Earth, the same size. The NASA Scientific Visualisation Studio has produced an animation showing this - click here to see it in your usual MP4 viewer.
Two final points though - why is the colour of the eclipsed Moon (and the halo) orange or red, and why is it not brighter than it actually is? The answer is that when light passes through the Earth's atmosphere it is not only refracted but also scattered and absorbed. Scattering affects light at the blue end of the spectrum more than at the red (which is why the sky is blue), meaning that by the time the Sun's light heads towards the eclipsed Moon only the red (and perhaps yellow) wavelengths are left. Also, because it has passed through a considerable depth of atmosphere it will have been severely attenuated. These two effects also account for the reddish hue of sunsets and for the fact that you can look directly at them without harming your eyes. Both effects are highly variable though, depending on the state of the atmosphere, and so the eclipsed Moon can look anything from pale orange to deep, dark, red.
While that is true for the totally eclipsed Moon, during the partial phase the margin of the Earth's shadow can sometimes have a blue or turquoise fringe (as shown by my pictures of the 2015 eclipse). How can this be possible if all the blue light has been scattered away? In fact this fringe is caused by ozone in the Earth's atmosphere, which absorbs red light leaving just the blue to violet wavelengths. It is seen on the margin of the shadow because it only exists in a layer high in the atmosphere, through which light refracts to form the outermost part of the shadow cone.