“No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President;” – Art. II, Sec. 1, Cl. 5
A distinction is made between a “citizen” and a “natural born citizen”, this much is self evident.
The purpose of the grandfather clause is to allow Revolution Era patriots born outside the country to be eligible for the Presidency.
Not just Revolution Era patriots born outside the country, but only those who were citizens at the time of the adoption of the Constitution are eligible.
So what citizens born outside the country would be citizens at the time of adoption? They are those:
I. Foreign born English subjects (however they came to be such) in the several Colonies who became US citizens by the naturalization power of the Declaration of Independance
II. Naturalized by one of the states while under the Articles of Confederation
English subjects naturalized by the Declaration of Independance includes those born in one of the Colonies, as well as those born someplace else as in Case I above. The place of birth is not a factor in their eligibility, it is their naturalization which makes them eligible.
There are natural born citizens, naturalized citizens, and aliens.
If the intended prohibition was “all citizens except naturalized citizens” Article II would have been worded so.
The time of the adoption is the only time naturalized citizens are eligible. They are citizens included in the “citizens at the time of adoption” clause – but they are naturalized citizens, as differentiated from natural born citizens.
Natural born citizens must obviously be born from citizens.
The meaning of Article II, Section 1, Clause 5 is self evident.
 United States Naturalization Law states that children of aliens naturalize upon the naturalization of their parents. Aside from aliens who else is in country?
Naturalized citizens, who are ineligible except at the time of adoption.
Citizens born from naturalized citizens and citizens born from native born citizens, both of whom are citizens born from citizens, ie natural born citizens.