It depends upon the context, Machjo. The preamble to the Charter was written quite recently so there you may be correct, in that God does not necessarily mean Christian God.
However, the preamble to American constitution was written several hundred years ago, in those days, there was only one God, the God of the Bible. So when US constitution mentions God, it very much means Christian God.
Just one of the ways in which Canadian constitution is superior to US constitution.
SJP, Robertus Ketenensis translated the Qur'an into Latin in 1143. A second Latin translation was issued by Ludovico Marracci, a confessor to Pope Innocent XI, in 1698.
If you read Gibbon's Decline and Fall, published between 1776 and 1788, he likewise shows a considerable knowledge of Islam for his time. If European scholars of those times had so much knowledge of Islam, and the founders of the US were educated people, I cannot imagine them having been so ignorant of at least some of the basic precepts of Islam.
The earliest translation of the Qur'an into English was that of Alexander Ross in 1649 based on a French translation, followed by Sale's academic translation in 1734 and Rodwell's literary translation in 1861, these last two being translated from the original. Many other translations have followed since. In fact, Rodwell's was motivated by an intent to provide an exegetical work refuting Islam. If you should buy a copy of his translation minus the footnotes, you'd never know he was anti-Muslim owing to his honest translation. But if you get a version with the footnotes, it's clear what his real intent was. I cannot imagine him going through so much effort except to try to counter perceived fears that some Britons at the time were attracted to Islam already.
Certainly if there were fears in the 1800's over Britons adopting Islam, then I can only conclude that knowledge of Islam had been spreading long before that at least in academic circles, which the literary works mentioned above clearly prove. Seeing that the founders of the US were among that academic elite, I cannot imagine them having been so ignorant of at least the most basic principles of Islam such as the oneness of God.
Let's consider too that many scholars of the 1700's, Gibbon included, were self-professed Deists. Again, if there were so many Deists at that time, I'm sure discussion of the subject of Deism must have preceded them long before the 1700s for it to have spread so much during that time. Let's not think the Europe of the 1600's and 1700s was some religious monolith. Deism was already a perceived threat to the established Church (as evidenced by critical responses to Gibbon's work by Christians who were offended by his critical analysis of the Christian community in the Roman Empire), and knowledge of Islam already widespread among scholars.
I can't imagine scholars in the US having been so ignorant of those trends, and not being aware that a belief in God was not the sole monopoy of Christians and Jews even in their time.
Let's not insult the intelligence of our forefathers.