The use of Creator was specifically not a reference to that Abrahamic God. It's the terminology of the Deists of the day, which basically believed in a creative force that kicked the universe off and then stood back and watched. It's basically just science, study of the natural world 'given to us by God'. They didn't have the Big Bang Theory and theoretical physics, it was contained in 'naturalism'. Deism is specifically a rejection of institutionalized religious thought that claimed monopoly on knowledge. Its heart is rationalism, as opposed to the doctrines of any religion.
It may not be a direct reference to the Abrahamic God, but it’s not an exclusion either. That’s part of the genius of the wording. Men like John Jay and Thomas Jefferson can arrive at the same place with the meaning of the text despite having a different picture in their head of the word “Creator.”