It is the same "word", in Aramaic, the word Jesus said would sound like "Allahim" with the "A" as in "bat", not "bot"
I don't think God faults people for saying it either way or simply "God"
Maybe "the Lord" is something to think about. It's the same as "Baal" and most people think of Him as an unjust taskmaster.
KJV has good and bad points. I think replacing "Yaweh" with "the Lord" was probably the worst of the bad points.
The "idea" of "Allah" vs. "God" or "Elohim" is different though than the words used. The idea that Islam and Christianity have about who God actually is, is a "little" different. Most of both do inwardly and mistakenly call Him a "unjust Lord" instead of a "Savior" though.