Just for clarification:
The C lydian is just the notes of the G ionian but you start on C and end at C? There are no other changes to the scale pattern?
Ok, so modes are often made out to be extremely complex, more so than I think they justify, and maybe some historical context would help here.
The Greeks gave us the diatonic (seven note) scale. I'm going out on a limb here, but I'm assuming that, back then, instruments were only capable of producing
just those seven notes. So, music was written in that one key. If you wanted to play in a different "key," the order of those intervals would have to change. The Greeks eventually named them all based on, if memory serves, Greek tribes - Ionian, Dorian, Phrygian, Lydian, Mixolydian, and Locrian. You were changing the "key" of the music from a C to, say, a G, but you were also changing the "tonality" and the relationship of the pitches to each other. Music theory class was a LONG time ago, though, so bear with me here.
Eventually we ended up with the twelve tone chromatic scale and you could play any scale or mode from any of those starting pitches, which kind of brings us to the conversation we're having here.
So, I guess I'd say there's two ways to look at modes:
1) their derrivation. This is what we're talking about above, and to your point, yes, if you take the notes of a G major scale (I'm being careful here to NOT call it "a G major scale" but rather just the same notes), and play them from the 4th, C, from C to C but playing the notes you would in a G major scale, then yes, if the overall "key" you're playing them against is a C tonic, then you're playing C lydian.
2) their enharmonic construction. This is how they're usually described, which in this case, would be "C Lydian is a major scale with a raised 4th."
Note that both descriptions refer to the same series of notes, and get you to the same place.
I think the former is easier
to comprehend, especially initially, but you probably want to think about them both ways, as the later is probably a better indicator of how to use them harmonically. It tells you, basically, "what are the important notes in this scale?" Primarily, that would be the C major triad - C, E, and G - as well as the raised 4th, F# (which, in passing, is the major 7th of a G major scale). So, if you wanted to improvise in C Lydian against a C drone or a simple, say, C-G chord vamp, if you really wanted to drive home that "lydian" sound those are the notes you'd want to really be focusing on. If you were playing against a C drone and were just playing G major licks and ending a lot of lines on G, then you would technically be playing in C lydian, but the way you were phrasing wouldn't really necessarily capture much of the color of that mode.