I don`t think you can just go off the spelling, when we went to Zakynthos, we stayed at ALIKES, spelt that way.
Αλυκες is the real spelling so the confusion comes in only because of the need to make it readable for non Greeks. I don't think really one spelling is any more correct than the other. The third letter, upsilon, actually translates as a "u" but a capital upsilon in Greek script looks just like a "Y" and is used as a "Y" in Greek when it comes between consonants. The pronunciation for upsilon though is "i" in modern Greek but "y" in ancient Greek. There is no real equivalent to "y" in the Greek alphabet as far as I can tell but "i" is "ι" or iota. So should it really be Alukes ?
Having said all that though 99% of the official Zakynthos maps and road signs obviously use ancient greek as they all say Alykes
Hope that's cleared it up for you