"When he is thirsty, he drinks even water."
Translation:Když má žízeň, pije i vodu.
I can understand that. But I think the author wanted to stress that in the Czech sentence the "i" binds with "water", not with "pije". So he will resort to drinking even water and not some other drink. While in "he even drinks water" he will resort to drinking water instead of doing something else.
I understand the distinction you are able to make in Czech by changing the position of "i" in this case, but in English we would almost never say "...he drinks even water", but rather say "...he even drinks water" and use voice stress, another qualifier, or other context to explain whether he would drink something besides water, or do something besides drink the water.
As the English is now, it sounds awkward. I'm not saying it's grammatically impossible, just really irregular and a confusing example for non-native speakers.