The first thing is learning to hear the unique “colour” the #4 with the IV chord (F# with Cmaj) and the b7 with the V7 chord (C with D7 chord) bring to those chords.
Next thing is learning how the create tension and release with them, and listening out for subtle ways (and not so subtle ways) they fit into the melody/bass line/harmonies of songs you’re listening to.
Another mode debate
I would agree with this as one useful way of thinking and applying knowledge of modes.
If it simply says C and that’s the 4 of G, there would be an F#(raised 4 of C). Likewise if it says simply D and still in the key of G, then you’d play a C natural (lowered 7).
The most important thing I’ve come across with modes is to hear those color differences (i.e. raised 4 and lowered 7, yet both major chords) so you can identify the sound of a passage, chord etc, as quickly as possible.
For example “uptown funk”: there are
several times when the color of the raised 6 sticks out in lines that are played. I hear it’s a minor jam so I know it’s for the most part a Dorian jam.
How many James Brown guitar parts have that color....
Once your ear identifies the different sound of each mode quickly I’d stop thinking about any other way of using it and let it happen however it does.
The most use I get out of modes is to quickly identify a passage or chord. If you’re at the point where your ear tells you that sound on the fly, you probably know the fingering by then!
That said: the veracity of a chart is also up for grabs.
So in your example if it says C, use your ears to see if indeed it has a raised 4 (the melody or chord maybe)...very often it won’t. It’s entirely possible that the chords played are all Mixolydian for example, blues anyone? But your ear will tell you that if you work on the “sound or color” of each mode.
Leave Phrygian and Locrian for last, unless you’re in a metal band!