One more question about Russell Lydian theory. Russell uses as an example the Lydian C scale: C-D-E-F#-G-A-B-C comparing it with C maj scale, then he starts to explain his theory as if they belong to same key, otherwise why comparing C Lidyan with C maj? This often happens with jazz theory, which confuses me as I was trained approaching harmony through keys (we actually use "tonality" as term - we gives key different meaning, such as bass key, baritone key, tenor key, etc). 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!). Sometimes I also find theories in which Cmaj is related to C- as parallel and I find it quite odd, as there is not a direct relationship, as far as I know. Of course in music everything could be related, mostly depends on the sound that comes out, even more on who plays it. Thanks to anybody who will be available to answer.
If I understand correctly, you are asking whether C Lydian "belongs" to G major or "belongs" to C major. I would say neither, or both, depending on what you mean by "belongs". C Lydian uses the same notes as G major but the same tonic as C major. Harmonically and melodically, though, it's different from both.
I think this is just a confusion about terminology.
"Lydian Chromatic" comes from a musical world concerned with methods of scalic improvisation. Take what you will from it, but don't expect a fully-thought-out theory of harmony! It isn't a religion, to be followed in its entirety, just some concepts that may illuminate your wider musical journey to enlightenment.