Your citation from the Koran does not uphold your assertion. Your interpretation that since Ilah is a common noun, Allah is a name doesn’t work. Given Arabic grammar, you should read it as, “There is no god (Ilah) except The God (Allah),” since Allah is a contraction of “the (Al)” and “God” (Ilah).
Given that Allah means “the god,” not “a god,” it makes perfect sense that Allah would be used only for the Mohammedan god as the Encyclopedia of Religion and Ethics states. To use it to refer to any other being would be to acknowledge that such another being actually is a god. True, in the West, we might refer to a god in such a way, but Islam is much more militated against foreign religions taking hold than we are. We’ve dropped the formalities, hence we say “Jesus,” instead of “Our Lord Jesus Christ.” or Catholics say “Mary” nowadays instead of “The Blessed Virgin Mary.” Contrarily, we also say “god” when we mean “false god.” This was once justified in writing, where “god” could be distinguished from God, but now we even merge the two in speech.
On the other hand: It would seem troubling to me that Allah was previously used for a moon god, and yet the Muslims make no distinction. You’d think they’d want to clarify which god they were referring to, especially given the formalisms that surround Mohammed. They won’t say the name Mohammed without “(p.b.u.h.)” so they honor Mohammed, not some othe of the hundred million Mohammeds in the Muslim world. Yet there is no such honor afforded God himself. It’s as if despite their claims that Mohammed is only a man, he is exalted higher than Allah.