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.
Maybe this link will help you further:
https://www.mcclimon.org/blog/reconceptualizing_the_lydian_chromatic_concept/
To complete the confusion we could say (instead of saying C lydian is the mode of the 4th degree of the major scale=Ionian):
Each mode has its own row of halfsteps and wholesteps and its root tone is its 1st degree. This is historical more correct! But referring all modes to the major scale - as exemple preferring C major - is for students educated in the system of the western major/minor tonality as a pedagogical approach probably better understandable.
So we have to differentiate whether we think and discuss within the theory of Russell (where CDEF#GABC is mode I ("major") or try to explain it to someone who comes from the traditional Western theory (where Dorian C is mode IV of the G-key.)