Translation:Please cut off half of the cucumber.
"Cut the cucumber in half" is normal American English. "Cut off half" is not normal AE.
Does this mean take the cucumber and cut it in half, or is it take the cucumber and cut up half of it, or is it ambiguous?
This: "take the cucumber and cut up half of it"
"take the cucumber and cut it in half" would be "Разрежь, пожалуйста, огурец пополам (adverb)"
- отреза́ть (perfective is отре́зать) is about cutting and taking the cut part away
- разреза́ть (perfective is разре́зать) is about cutting itself
This sentence uses the perfective verb.
Отрезать половину огурца - разрезать его пополам.
Нарезать половину огурца - разрезать половину на несколько частей.
The natural answer to this (rejected) would be "Cut/slice the cucumber in half, please"
Yes. In usual American English idiom, the thing you "cut off" is smaller than what remains. If you cut something into halves, you "cut it in half" you don't "cut off half". If the thing you want is bigger than half, you simply ask for the amount you want - "Cut 3/4 of the [thing} for me". And you don't cut off the smaller portion and ask for the larger one. But, it's English, so there are always exceptions.
"please cut off half the cucumber" was rejected. Omitting "of" in this case is very common English usage, though technically incorrect. I think it should be accepted.
Still rejected: 30.09.2017. Incidentally, "half a cucumber" is perfect English. There is nothing "technically incorrect" about it. No one says, for example, "half of a dozen" or "half of a sandwich." The word "of" is superfluous and any good editor would delete it.