First, I'm sure most mexicans in Houston definitely agree with that. The best mexican food in the city is from either their mom, grandma, or an aunt.
The deepest history I've ever found for the burrito is that it comes from northern Sonora, but it's shunned by most of Mexico as an american food. So, I guess one would consider it an adopted american-mexican food. The reason it was invented and it's namesake are both obvious.
Burrito offsprings - The chimichanga is from Tucson, and the "mission" burrito, aka "california burrito", is from SF.
Fajitas are from Texas, it became popular after being given as payment to cowhands and vaqueros. Through most of history, the skirt steak was considered a very undesirable cut of meat. In Mexico, they are called Arracheras. They were very tough and not very juicy, so only the poorest families bought the skirt steak. Modern Arrachera is usually made with fajita, not arrrachera.
Carne Asada, Mexico's version of barbecue(Barbacoa is not barbecue it's roasted cow head, and Barbecue refers to american/texas barbecue) and comes from Tamaulipas(north and central). Normal ingredients are the meat, oranges, beer, and onions. It is usually served with corn tortillas and is usually accompanied with beer, guacamole, totopitos, and frijoles a la charra. The dish "Carne Asada(a la Tampique