Can you spread the different herpes to different body parts?

I got a cold sore. My friend and I were discussing the ramifications. She thinks that if you give someone oral sex while having a cold sore, you will give them genital herpes. And vice versa.

I don’t think that sounds quite right. Since Herpes is a lifetime virus, when it re-occurs, it’ll cause blisters in the area that is specific to its genome. So, I thought that if person A has oral herpes and gives person B oral sex, person B may develop sores in the genital region but in time, the disease will re-emerge as oral herpes.

Does that sound right or am I completely off? It’s been a while since I studied viruses.