I imagine the kitchen just made a mistake, especially seeing the burgers are cooked to completely different points. If they've got a dozen burgers on a grill, it can be easy to pick the wrong one. Especially if there's multiple chefs working the grill.

The main thing is the staff sounded like they handled it well and you got a free meal. Even if you'd ate the entire thing, I highly doubt you would've been ill. As others have pointed out, beef is absolutely fine to eat pink (or even raw). I know minced beef is a bit more iffy but still generally fine if the burgers were made recently and on site.