I am studying Cold Play's "Don't Panic", and in the piano arrangement, the arranger has changed a major6 chord to an inverted chord with the same bass. Here is the score
In the original recording of the song the chord is a G6 chord played on a guitar (about 0:50), so why would the arranger put that C there for the piano arrangement on that particular note? The arranger could have kept that B throughout the whole bar yet he/she decided to use C and change it halfway through the bar. Why?
If I follow your question's concern, I think you are expecting the chords symbols to be...
C/G then G6 or
Gsus4 then G6
...because of the C in middle staff, first beat.
But the chord symbols are there to show what a guitarist should play rather than some kind non chord tone label or analysis.
The chord in this section is G6 and their is a non-chord tone C resolving to a B.
Another way to think about this situation is a very old idea from figured bass harmony. A chord in second in version like C/G is a harmonic idea, basically dealing with chord roots. But figured bass looked at it as a double appoggiatura over a G bass, it look at it as a contrapuntal thing, it would be called a "chord of the sixth", but the G bass was the fundamental thing rather than a theoretical root of C.
From the figured bass, contrapuntal perspective the C is an embellishment to resolve, not really a proper chord tone.
If you go with the modern, harmonic, chord root perspective you should be identifying 6/4 chords in the four categories: cadential, passing, pedal, or arpeggiated. The example doesn't include before/after one chord so we can't really explore it this way.
This seem like a lot of theory to throw around. I think the simple idea is don't casually identify 6/4 chords. First suspect them as non-chord tone, contrapuntal elements, unless you can fit them clearly into one of the four harmonic categories.
As far as I can see, the arranger has written the following:
I don't recall any rule that states that if a guitarist plays a G6 chord, then a pianist absolutely must not play a C note, even for a fraction of a second.
Chord symbols are not a harmonic analysis: What is the relation of guitar chords to actual notes in the measure?
It would be unrealistic to demand the C note to be reflected in the accompaniment chord symbols. It is perfectly fine to have a backing band play a chord - however they feel appropriate - and write other notes for other instruments. For example, for harmonizing a melody with a fixed sixth or third.
Edit. From the OP's comment I see that the real question is: why wasn't the arranger faithful to the original recording, which only has a guitar in that part, and the guitar does not play a C note. Well, who knows why that is, but it's an arranger's job and even duty to make some kind of a creative contribution. Without further information I have to assume that the arranger liked the sound of the added C note, and thought that it felt appropriate to have a piano play it at that point.
Edit2. After listening to the recording posted by Laurence, on the first "yeah" word, the lead guitar and the melody together play a distinct C chord, and the bass plays a G. The acoustic guitar in the background may or may not play a C on the first "yeah", it sounds like muted notes to me. The G or G6 chord, with a B note, is played only on the "do" word. The transcription or arrangement is OK, so your claim that it's not faithful to the recording must be based on a different version of the song, or something.
Listen to the track. This phrase occurs at 0'50". I'm sure I hear two different chords. Which means the piano part is correct. The chord symbol is wrong. Or, if you want to be polite, an over-simplification.
Now, what the actual intention was is another matter! When that section returns near the end of the song the guitar plays something a bit different. Maybe deep artistic intention, maybe just sloppiness.
First, I guess the author meant "... in particular that CHORD...", instead of "... that NOTE...". Second, the chord IS ACTUALLY a CMaj⁹ chord. It's easier for a guitarist (guessing he's not proficient in harmony) to place the fingering shape of a G⁶ chord, finger 2 on 3rd fret. However, for a full PIANO part, and much more if to improvise, the note "c" should be present to understand the SOUND of the moment. (I play professionally both jazz guitar and piano, and am an arranger also).
