"Natural born citizen" doesn't mean "born on US soil" it means "born a US citizen". Which means anyone born on US soil (domestic or overseas, since military bases and embassies are US soil), or anyone born to US parents. Yes, of course this could include Obama by any stretch of the imagination, since literally nobody disputes his mother's citizenship.

Now, Cruz is disqualified since you can't be a natural born citizen if you're not born, you're a collection of snakes and spiders that have been gathered by dark magic and stuffed into a skinsuit.