They use the word "Allah," for God, as that is the proper Arabic term for the Abrahamic one true God....
But you don't get to redefine words that have accepted meanings.
Assuming you're not an Arabic Christian, as long as you don't use secondary meanings for foreign words when both the primary meaning is far more restrictive and controversial, and there are other words in English that fulfill that secondary meaning, there won't be any misunderstandings.
You are entirely free to reject Muslims' belief that their Allah is the same deity as our God. I'm not particularly sure of that myself.
Actually, I am quite sure of that. Even though Muslims claim that Christians (and Jews) are of the Book, the fact that they reject both the deity of Jesus Christ and the Trinity of God means that, by their admission, they don't worship the same God as Christians.
Your position is internally consistent.
Just curious. Do you apply the same logic to Jews, and deny that they worship the same God as Christians? They deny the deity of Christ and the triune nature of God exactly as Muslims do.
BTW, quite a few groups that claim to be Christian disagree with one or both of these beliefs, or differ significantly with your (likely) definition. Do you deny that these groups are Christians?
We're arguing over symantics though we all agree on the one thing -- the Moongod the slammies worship is the Devil.