The lydian scale is built on Fa (FaSoLaTiDoReMiFa=WWWHWWH). In the key of C this is F-G-A-BC-D-EF, in the key of G this is C-D-E-F#G-A-BC. 

> So, to me C Lydian belongs to the key of G (1 sharp). In fact, comparing the two modes, Lydian and Maj, he builds his theory on F of Cmaj and F# of Lydian (of course!).

You are correct: 

Lydian C is the IV. mode of the G major scale but quite different (melodically and harmonically) from the key of G major or C maj.