The link you provided has no information as to whether that's a correct etymology or not, only that 月食 yuèshí is made up of characters 月 yuè 'moon' and 食 shí 'eat'. There are numerous other explanations for why those two characters could be used: for example, perhaps 食 used to have a different meaning which dropped out of common usage, or maybe 月食 used to be written with different characters and people switched to the easier-to-write 食 from a more complicated character, or any of a number of other reasons.

I don't know which of these is true, if any of them is. But your link doesn't have enough etymological information to indicate one way or another, either!