You're right that it's not exactly complete. Time-translation symmetry is what would make an experiment "not depend on when the experiment took place," but time-reversal symmetry, another "time symmetry," would mean that it did not matter whether a VCR tape of the experiment was playing forwards or in reverse, so that both directions appeared to be completely reasonable events. (For example a car driving forwards is just as reasonable as a car driving in reverse.) Time translation symmetry appears in real life, but time reversal in general does not.