Quite right about the piece itself establishing the tonality/modality. But "key signature" might be a slippery term here, the reason being that that really refers to writing, and many (most) people may not bother to write modal music with specific "mode signatures." IME they more often tend to use the key signatures of the nearest major/minor key when they write. Thus, a tune in C mixolydian might be written in C major, with Bb accidentals added as needed, and the tune might be referred to as being "in C." In addition, the key signature never establishes the tonality/modality, it only reflects it to a greater or lesser degree. I think maybe what you mean is something like "note complement" or "set of tones"?