A token alternative answer from an erstwhile GDR border-crosser at Grenz*übergangs*stelle Friedrichstraße in Berlin. Hence avoidance of any echoes of the past.

This may help by making it a little more general.

Given that the primary purpose of a GÜB is to provide proof that a visitor has left Germany, I find "border crossing certificate" a tad inadequate as that doesn't indicate the direction of the crossing.
(Note: I was a liaison officer on the German/Czech border during the Cold War and we dealt with a lot of "border crossers", all of whom crossed INTO Germany. That may influence my thinking.)