Let me try something here: I'm going to make a series of assertions, non-argumentative in nature, and some or all of you tell me, or the thread generally, where I am wrong.
1. ilah is the Arabic word for god. Note the lowercase. If we were having a peaceful conversation with a Muslim about Zeus or Thor, we would agree that each is an example of a false ilah. Of course, it also means god when speaking of the real god.
2. Allah means God, and is most likely a contraction of Al- (the) plus ilah, in the same sense as we can talk about Lord North or Lord Salisbury in British history but The Lord means God Almighty.
3. Islam uses Allah as the name and/or title for the God whom Mohammed taught about and whom they proclaim.
4. Arabic Christians use Allah where we would use "God" as the proper title for the Holy Trinity.
5. There are allegations, which I personally have never seen substantiated anywhere but anti-Islam hate sites, that Allah was used as the name or title of a moon god before Mohammed when the Arabs were polytheistic. I'm not saying whether that's right or wrong, just that I've never seen it borne out on a site with documentation I feel I can trust.
6. Mohammed believed himself called to proclaim the God of Abraham as the one true God, in place of the pre-Islamic Arabic pantheon. He used Allah, "the God," as the name or title of the God of Abraham.
7. His conception of who God is, is entirely at odds with the Christian conception, and in large measure with the Jewish conception too. And obvously it does not include God as the Holy Trinity, among many other errors.
8. Nonetheless, because he was pointing to the God in whom we believe, the God of Abraham, Isaac, and Jacob, of Moses, of the Prophets, the one of whom our Lord Jesus Christ spoke and whom the Apostles proclaimed, we would be in error to call Allah a false god. A very erroneous understanding of God, definitely. But Mohammed was pointing to the right God, but saying completely false things about Him.
Does any part of that, spelled out in sequence, seem in error to anyone?