According to your Wiki, the lunar eclipse is caused by a "shadow object."
The Lunar Eclipse is red because the light of the sun is shining through the edges of the Shadow Object which passes between the sun and moon during a Lunar Eclipse. The red tint occurs because the outer layers of the Shadow Object are not sufficiently dense. The Sun's light is powerful enough to shine through the outer layers of the Shadow Object, just as a flashlight is powerful enough to shine through your hand when you put it right up against your palm.
The globular earth is said have a circumference of 24,900 miles while the atmosphere is said to extend only 100 miles around it. If the RE model were true, and the redness of the shadow was caused by the sun's light filtering through the earths atmosphere, then the earth's shadow upon the moon would only have a slight sliver of red around the shadow's edges. The moon could not turn entirely red as it does in the above image. The fact that the moon turns entirely red during a Lunar Eclipse can only suggest that the light of the sun is flowing through the majority of the body which intersects the path of light. Clearly an impossibility in the RE explanation.
This explanation is chock-full of flaws. I'll start with the second paragraph, which says that the shadow should only be slightly red at the edges. This is based on the assumption that Earth's shadow is projected onto the moon, which is wrong. Sunlight is refracted through the atmosphere onto the moon's surface. Think of how you can see the light from a sunset for quite a while after it goes down.
Now about the "shadow object." (This is the part where I have to suppress the urge to punch my screen.) Does it not seem odd that this object would need to have orbital parameters similar to that of the earth itself, and just so happen to only cast a shadow on the moon when it's full? Where is this object during solar eclipses? If it's the thing casting a shadow instead of the moon, why don't we see the same translucency as during a lunar eclipse?
The depressing thing is that someone actually came up with this, and others actually believe it.