When I refer to God or religion I usually just mean the traditional conception.
Nature of God has been in debate not only between Christian and non-Christian, but also intra-Christian factions. From the beginning of the Christian sect, they divided themselves between the "Judiazers" and "Hellenizers" relating to the nature of their Messiah. Then the infinite variations of Christianity follows: Arians vs. Niceans, Nestorian vs Ephisian, Coptic vs. Chalcedonian, Catholic vs. Orthodox. The division between Western and Eastern Christianity must also be mentioned, with the West's penchant for legalistic doctrines vs. East's penchant for revelatory dogmas.
Of course, the so-called Reformation unleashed infinite factions baying for each others' blood over the nature of their one god - lower Protestantism can't be termed triune god worshippers as their theology is but a garbage dump of rejected ideas. I don't know much about the Old World, but here in the US, many "Christians" believe that Jesus was a 'Murican who established his church in Texas and gave us the King James' Bible.
And isn't all that a kind of (maybe weak) argument against God's existence, seeing how it's rather obvious from all this that it's men creating God rather than the other way around?
:-). I like the way you challenge theists to refine their world-view. What is the nature of man? As with the nature of God, it seems that the definition regarding the nature of man also has infinite variations. Some, our "he shall not be named" (maybe anomen?), posit that man is just a meat-bag of chemicals, dancing to the drumbeat of instinct; others, like Dawkins, believe that man is a meat-bag of chemicals, gifted with the ability to transcend his instinct; others, like Plato, thought that man is amalgamation of meat and soul, and the soul defines man; Descartes considered that the nature of man is circumscribed to only his thoughts, as nothing in reality is certain. Do all these varied perspectives regarding the nature of man negate that man has a nature, or is man also but an illusion?