The whole idea that you can be "dating" and not be "bf/gf" from the OP is one that is alien to the British experience.
In this country, you can hang out with someone (in a group, at schoolwork etc) but doing something just the two of you ("a date") makes you a couple - albeit a temporary one. It is not normal in my experience to "date" more than one partner at a time in Britain.
Having explained that, you will realise that for an American to say "I'll wait until we're bf/gf" is irrelevant to the British dating experience, because there is a transition in American culture from "dating" to "going steady" which does not exist here - we have implied monogamy from the first date.
Many British people will have sex on a first date, and many will wait months and months.
I have never "dated" someone I hadn't already got to know and like socially, but I have generally slept with them on the first chance I got
With my wife, we met online, talked for a while, phoned and exchanged pictures, and I invited her to meet me in person at an art gallery. I was fully expecting to go home after, but she invited me to dinner and I stayed over. As others have said - don't plan too much.