You can't run for President in the US if you were not born in the US, even if you are American.
There are residence requirements for foreign-born US citizens to pass on citizenship to children born abroad, which already contradicts your statement. This applies to mothers and fathers (with a paternity test if not married). Furthermore, it hasn't been legally settled whether someone born a US citizen abroad counts in this clause, though personally I suspect anyone born a US citizen (not just, and not always, children of US mothers) would qualify if challenged.