Because "God" is a proper noun, denoting the Christian God, while "Nature's God" or the god of nature, refers to any God who might exist. It's a gesture toward inclusion of people whose gods might not be referred to as "God".
I'm curious as to what evidence you have that the Founders interpreted it that way, simply because you, over two hundred years later, interpret it that way.
They really didn't avoid endorsing Christianity, and certainly not to be inclusive to non-Christian religions, because those virtually did not exist in the US at that time. What they were trying to avoid was a federal endorsement of any SPECIFIC Christian group, primarily because they didn't want the federal government interfering in the individual STATES endorsing specific churches. You do know that there were several states which had official churches at that time, don't you? From the perspective of the people of the US in 1787, "different religions" meant Baptist, Episcopalian, and Catholic, not Christian, Buddhist, and Muslim.
Yes, I know about the state endorsed churches, since more than half the states were essentially founded by different sects seeking to go off and form their own communities. And the Treaty of Tripoli would seem to suggest that the founders didn't think our nation had any problem with Muslims, or "musselmen" as they called them at the time.
America is a nation composed primarily of Christians. It is also a nation composed primarily of Caucasians. But that does not make America a "Christian nation" any more than it makes America a "Caucasian nation".
Well, actually, it does.