I used to think that English deer = German Reh.
However, apparently English deer = German Hirsch, and German Reh is specifically English roe deer. (English red deer, for example, is a German Rothirsch.)
Ah well; I was never very good at telling apart all those antlery animals in the first place, in any language.
Then you have elk meaning something different on both sides of the Atlantic: in North America it's a German Wapiti, while in Europe it's what Americans call a moose and Germans an Elch. Confusion!