The mistake comes from reading Emmerich de Vattel, who was often cited by the Founders, but Vattel was Swiss. He was describing the legal doctrine of the European Continent, where the rule for citizenship was jus sanguinis, or that citizenship was based on the citizenship of one's parents. That was not the doctrine of Anglo-American law, for which the rule was jus solis, or that citizenship was based on the soil of the nation, the location of one's birth, and not on the citizenship of parents. It is discussed in Blackstone, and particularly in the footnotes of Tucker's edition of Blackstone (1803).
"Natural" in that context means "according to natural law", in other words, by the rules of the common law then established, which pretended to divine natural law in the ways court decisions were made.
It is a common mistake to think that "natural-born" is synonymous with "citizen at birth". One can also be naturalized at birth by statute. That was done for John McCain, retroactively by a statute passed after he was born on the soil of the Panama Canal Zone, which was a leasehold, and not U.S. soil. McCain was definitely not "natural-born", no matter what a Senate resolution might say.
So Obama is eligible only if we was born on U.S. soil, which could have been in the airspace above U.S. soil, or in U.S. coastal waters, or on the grounds of a U.S. embassy abroad (U.S. soil by treaty), or perhaps on a U.S. flag vessel at sea in international waters. Not if born in the air over international waters if it was not a U.S. flag aircraft, and not if in the airspace over a foreign country.
All this is actually spelled out with some clarity in U.S. code, which at least on this subject happens to get it mostly right, although it is ambiguous because it doesn't address "natural born" at all. Some people manage to mis-read 8 USC 1401, but that section is not defining "natural born". It is defining "citizen at birth", and it is part of Chapter 12, Subchapter III, Part I, titled Nationality at Birth and Collective Naturalization. Only the first two clauses are about natural birth. The rest are about collective naturalization.
A birth certificate is evidence of location and date of birth, and is usually dispositive. However, a "certificate of live birth", which does not show location and date, is not. That is all we seem to have for Obama, and what has been offered is clearly a forgery. More dispositive would be witnesses to the birth. Unless the mother was alone when it happened, there should be some witness, even if no longer alive, who made a record of the event.
The burden of proof of citizenship for purposes of removing an individual from U.S. soil is on the government. However, the burden of proof for purposes of voting or holding office is on the would-be voter or officeholder. Even if the candidate was actually born on U.S. soil, if he can't prove that he should not be deemed eligible. It doesn't have to be proof beyond a reasonable doubt. Likely few of us could offer that. But there does need to be a preponderance of evidence.
There have been some efforts to exclude Obama from the ballot until he proves his eligibility, but it is useless to try to exclude Obama from the ballot, because it is not Obama that voters vote for. They vote for electors pledged to vote for him. The ballot might show the person to whom they are pledged, but that pledge has no legal status. The same people could be said to be pledged to vote for Mickey Mouse, when everyone knows that Mickey Mouse is a pseudonym for Obama. Or to some real person. Nothing prevents them from voting for anyone else when the time comes.
The only points at which an ineligible candidate might be blocked is (1) the counting of the elector's votes; (2) the certification of the elector's votes by Congress; and (3) inauguration. None of those points are subject to the orders of a court. For each of them, enforcement depends on the people involved voluntarily following the Constitution. Nothing can make them if they choose not to do so.