We don't have good studies to know if you are likely to have it both orally and genitally from the encounter that resulted in you contracting hsv1 genitally. The few studies we do have only looked at the appearance of obvious cold sores as an indicator of infection with hsv1 orally and that ranges from 1/4 -2/3's of folks who contracted hsv1 genitally, also contracted it orally around the same time. Since the vast majority of folks who have hsv1 orally don't get obvious cold sores to know that they are infected anyways, it's not a complete picture yet.
So what do you do? Well in your current relationship it's not an issue anyways. If in the future you move on to another relationship, you'll want your partner tested for herpes anyways before becoming intimate. If they already have hsv1 ( which most adults do ) it won't be an issue. If they have hsv2, then you'll want to take precautions to reduce your risk of contracting hsv2 from them. If they are completely hsv negative, then the two of you can discuss then what precautions you want to take for what then.
Genital herpes is always genital herpes regardless of type. You can have either hsv1 genitally or hsv2 genitally. It doesn't automatically become hsv2 because it's below the waist or anything like that.
Hope that helps somewhat with your questions even though part of what you've asked, we really can't give you a good answer for unless you get an obvious cold sore to know that yes indeed you do also have hsv1 orally.
grace
I contracted hsv1 genitally from someone who was having an outbreak of oral herpes .. in the 3 years Ive had it, I've only had 1 outbreak since my original. I take Valtrex, probably unnecessarily, but it give me a piece of mind on the transmission factor. so YES, you can have genital herpes 1 and not have 2.
If you have genital hsv1 then it's impossible for you to have it on your lips. Once you get it one place it stays in that one place.
But for the record 90% of the population has oral herpes and only about 20% show the classic cold sores. So basically you can kiss whoever you want, as virtually every person has hsv1, usually on their mouth.
I only advise you tell your partner to wear a condom for sex.