I think that in English the phrase "177 contains the same digits as 117" is, at the very least, not obviously false? "The digits contained in 177 are 1 and 7."; "177 is made using the digits 1 and 7."

The same: same kind of digits, same amount of each of them. Technically, yes, it's a permutation. I think if it were like you mentioned first then that should be explicitly mentioned and because the same means what I said, explaining it is not needed.

The specification "contains the same digits" seems ambiguous to me in the case where a number has repeated digits. For example, 1272: is the solution 1271 ("contains the same digits") or 1227 ("is a permutation of the input digits")?