Consider any one segment of a 7-segment LED display. Say that two different people?

Yes, the circuits could be different but logically identical. One may be better than the other (e.g. uses fewer components or faster in responding to inputs etc)

(continue) each design their own version of a circuit to control that segment. They agree on which input bit patterns (patterns of T and F) represent which of the inputs 0-9. They also agree on when the LED should turn on given each of those inputs. However, the circuits they produce use very different gates. Could two such different circuits both be correct? If so, can the circuits be logically equivalent? If so, must the circuits be logically equivalent?