Yes, both types can spread to either place, and more places. The main difference is that type 1 is usually associated with the mouth and cold sores there, and type 2 is associated with genital herpes in that area. However, type 2 can spread to the mouth and type one can spread to the genitals. Type one is usually less aggressive in a genital form than type 2 is.
If a person has a cold sore on their mouth and they touch that to someones genitals or eye or mouth then it can spread, vise verse.
As for your doctor, you should get the results, that way you will know.