If that were true then we wouldn't be seeing all these contentious threads. The fact is that sources as diverse as William Blackstone and James Madison and Roger Taney as well as other lawyers and politicians since them have not defined natural born citizen as you and Vattel do. Why are you right and they wrong?
Let us reason.. shall we? Can we agree that citizenship and natural born citizenship are two different things? Of course they are, the Framers would not embellish the Constitution with meaningless phrases. Agree? So, citizenship is DIFFERENT than natural born citizenship in some fashion. Agree? What could the difference be? Or, better put, what would make natural born special? There are only a few options. One would be that both parents are citizens. Why? Because if one parent is a citizen and one is not a statute is absolutely necessary to assure American citizenship. Why? Because other countries have their laws, and their laws may confer, or even demand citizenship if one of their citizens is the parent. So one parent citizen is nothing special, it doesn’t even guarantee citizenship. The only other thing to consider is soil. Soil is not typically the deciding factor, because we are dealing is SOCIAL compacts. But the Framers understood that soil IS important to the PERSON. An example; my son was born in Michigan, but left when he was one year old. To this day he is a fan of Michigan basketball BECAUSE HE WAS BORN THERE. So soil is important because it confers patriotic feelings. This is precisely why the Framers wanted soil. They wanted the President to be grounded completely by strong patriotic desire. There is only one kind of citizenship that guarantees patriotism... Natural born. Born of parent citizens on the soil of America.