There is a vocal contingent of Freepers who agree with this definition, however, as the article state "Natural Born Citizen" is not defined in the COnstitution or the US Code, so definitions such are yours are really just theories at this point.
If that *were* the standard then Obama would clearly not be eligible for President: his father was not "US Born". I am pretty sure that there were other Presidents who had parents who were not born in the USA, though they had citizenship at the time the future Presidents were born.
The definition you have offered is about the very most restrictive offered by any birther. It's not the rules we are operating under today, obviously.
I am 68 and yes, that was exactly what my parents always said natural born was. Both of my grandfathers said so also. My family on both sides was in the US before the American Revolution. I am a card carrying CAR member.
All four of my grandparents were in this country before it was the USA.