I'd use the first one Richard!! It's a bit like moveable Do (as in Do Re Mi) vs. fixed Do.

I can't see ANY reason to conflate modes and keys. It's clearest just to specify G (the key) and Mixolydian/Ionian/whatever (the mode).

The whole point of an identified mode is that it's a structure (or tonal colouring) which you can transpose into any key, regardless of what note you start on. So you get Mixolydian C by simply flatting the 7th (i.e. Bb) and Ionian G by raising it (F#). It's by far the simplest and clearest way, and keeps modes and scale pitches in separate camps, where they belong (..."if God had intended them to be mixed...").

DB, ya still with us? DB???   DB?????????????   :-)