Hi, thanks for your question. A cold sore is an outbreak of herpes simplex on your face. It is possible if have type 1 or type 2 or both on the face. Type 1 is more common on the face. It is possible to have type 1 or type 2 or both on the genitals. Type 2 is more common on the genitals. Most commonly cold sores are caused by type one.  So yes, you are right that a blood test would confirm herpes. That is why herpes testing is not routinely done at STI screenings as the majority of people will have a positive herpes test and the blood test does not tell you where the herpes is.
Herpes can be well transmitted from the mouth to the genitals and so then type 1 on the face becomes type 1 on the genitals.  If you have type 2 on your mouth and then have contact with the genitals, the herpes can transfer and be type 2. So just to be clear the type does not change with location but rather stays the same type.  To lessen the risk of transferring herpes from the mouth, consider having no contact from when you first feel the cold sore coming on to when it has completely healed.  Your partner can also consider using a condom to cover the shaft of the penis to prevent direct  contact with your mouth. Hope this helps. Health Nurse