maybe more of an american thing but pretty much any place you go to that's making a burger to order will ask how you want it done. I think it's just more a reference to the fact that a steak would look like this at medium rare as opposed to any adherence to correct culinary terminology.

I used to have a buddy that was a bartender at a place and he knew my penchant for rare steaks and all that. I'd occasionally order burgers from there and they'd cook it to be a bit pink inside than they were supposed to and I never got sick from it.