How did they used to keep track of making sure someone was born in this country and not an impostor, or some kind of "mole insert"?

Before the days of electricity I guess most specifically. when things could be fabricated on an any given man any given way basis..

i dunno about in the states, but over in Britain they still had census records long before electricity

Public electricity (late 1800s) didn't make anything more reliable. People just used common sense.

Martin Van Buren was the first American president who was actually born in the United States. All the other presidents before him were once former British subjects, back when America was still a colony to Britain.

The first US census was done in 1790, but it was understood at the time that until the first *actual* American became president, they couldn't exactly enforce the "must be a native citizen of the US" rule. It's also helpful to remember that excluding all the people born after the American Revolution concluded/America was recognized as a country, everyone else were former British subjects.

does that make us all invalid? We can do anything. is that a secret?

Communities were more tight knit back then. People would notice if someone new moved into town. Also, as stated above, a census would have been carried out. I don't understand why you think this is a "presidency" question. This isn't something limited to the US. Can't you remember why Mary and Joseph supposedly trekked all the way to Bethlehem? A census. Census' have been carried out for thousands of years to keep track of this sort of thing, and before aviation came along and made travel far easier, ships transporting new people would be inventoried and easy to check and land borders the only real problem. But once again, communities would keep that issue in check. It's really only the modern world with it's easy travel and open borders, large cities with millions of nameless faces, that requires the use of electronic registration.