While it is conceptually possible for you to have HSV 2 both orally and genitally, that is rare. But if you had it orally, you would shed very rarely. And yes, if you got HSV 2 first, you have virtually no chance of acquiring HSV 1 subsequently.
We know that 95% of the time, someone who has HSV 2, has it genitally. The other way for you to know if to have one of your outbreaks swab tested and typed.