I know what I am and that is Bisexual. However, I would only start liking someone after I get to know them first. I do not know what that makes me. I keep researching this term demisexual and it basically says that you really have to develop a deep bond before a relationship happens. So, Biromantic Demisexual? is that even a real name for it?
It's nice that you've identified how attraction usually works for you. If you feel the need to have something that specific, yes, biromantic demisexual is a name for it. (Besides, even if it wasn't a name, you could invent it.) You can try demi-bisexual on for size, see if you prefer that. Don't get too caught up in labels. You know how your attractions work, so you don't need other people to validate the labels you choose for yourself. However, choosing a more "complex" label will make it a bit harder to explain if you ID as that while coming out.