I explained in my post that Islam is defined in the Quran as the very continuation of teachings in Judaism and Christianity. Loosely summarized, the Quran says that Christians mistook Jesus to be divine.

The Quran does not say, "Everything before this is crap. Here's the truth." It says, "We continue the teachings of the previous faiths, but here are a few corrections to what they believe." Again, very loose generalization intended to give the big picture that Islam, as defined in Islam, is a continuation of Judaism and Christianity. It doesn't adopt everything 100% from those religions, but it is a continuation.

Thus, to say that the Judeo-Christian God is not the God of Islam is actually in opposition to the central teaching of Islam.

Whether the differences are irreconcilable or not would depend on the point of view. To Christians they are irreconcilable because the Quran disagrees with what Christianity says is the alpha and the omega. However, to Muslims much of Christianity (sans the Trinity) is an accepted and necessary precursor to the teachings of Islam.