As a Brit I'm a bit skeptical about anything below a well done burger. We were taught that you should always fully cook ground beef as the mincing process can mix in pathogens from the surface of the meat. I suppose this doesn't apply if the beef is safe to make tartar with or maybe if you sear the outside of the cut you're going to mince before mincing it.
That said a number of mid-range burger chains offer rare burgers here, and in such cases I tend to go medium which typically nets results on the less overcooked end of well done anyway.
New regulations came in a few years ago to cover that. Hopefully anywhere serving rare burgers is diligent about their supply chain and kitchen processes.
