Unlikely... however NOT impossible
It is possible but unlikely in my own book.
If she had herpes type 2 in her face, it would probably be a lot more than just a few cold sores.
In general, type 1 is at fault on the lips.
In fact, type 1 on the lips provides partial immunity to HSV type 2 for the sexual organs. That might be one of the reasons why she doesn't break out between her legs.
She will have to get her cold sores tested by a lab.
I have type 2 in my face and let me tell you that despite my good immune system, as soon as I eat certain foods, my face gets very itchy. I would be very surprised she has type 2 on her lips
Cold sores are a good thing (type 1) because as I said, they provide 'partial immunity' to the worse kind: herpes HSV-2.