I confess I have no idea what you're saying. I feel as if you're picking at nits. The quote was from wikipedia; Mel bay's book says: So G Mixolydian is indeed a mode formed from the major scale of C...which means to me the key of C. I'm not sure how to differentiate between the two as they share a key signature.
I mean...a series of white notes from G to G IS G Mixolydian by definition, isn't it? I don't see why saying "you're IN C major" makes that a false statement.
Or maybe you're just saying that C major the key and C major the scale are not the same thing?
This has been discussed a lot earlier in the thread. No, a scale and a key are not the same thing. One of the problems with the way we discuss modes is that we use the word mode for a couple of different things. A mode can be a scale but can also be a key (or "key," if you want to reserve the term key for standard major-minor keys).
A scale is just a set of notes associated with a key. It's only half of what you need to establish a key, the other part being the tonal center or tonic (or final, in the older modal terminology). The same set of notes with a different tonal center is a different key. Again, this was discussed quite a lot earlier in this thread, so you might want to review that.
No, G-G on the white keys is not a G mixolydian by definition, though that's one of the things it can be. And no, if you play a sequence of notes that looks like G mixolydian (mode = scale, just a set of notes), that doesn't automatically mean you're IN G mixolydian (mode = key.