So, I would like to clear up some of the confusion a bit!

In the picture are some stones from my own personal collection. True moonstone typically has a cat's eye effect, while rainbow moonstone (which is a white variant of labradorite) usually has a blue flash to it (it can have other colors or now flash at all depending on quality) with no cat's eye.

True moonstone is also usually a sort of milky color with a plastic-like luster when polished/cut (similar to opal) while rainbow moonstone looks more like glass.

True moonstone also comes in several different colors while rainbow moonstone is always white or clear (when it's grey or brown it's just called labradorite). I have seen dyed rainbow moonstone before, but naturally the only color is from the flash.

I do hope this clears up some of the confusion between the two types of stones.