The proposed definition of "natural born" is nothing more than a matter of conveniently crafted wording in the service of wishful thinking.
The entire birther thing is little more than desperate but lazy people, looking for a magic solution to a problem that really just requires hard work, patience, and a rational and focused political organization.
Correct.
And the appeal to Vattel, as I've pointed out, is problematic as well.
Vattel's system is markedly different from ours: the children of non-citizen permanent residents are not citizens under Vattel, but are clearly US citizens under US law.
He himself points out that there are significant real-life exceptions to his ideal definitions of terms.