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Do you know of any good resources for learning how this works? Things like the role a certain chord plays within a piece.
Jazz functional harmony and classical counterpoint would be two areas of study that would get you insight on how to answer stuff like this.
For this specifically, I'd say more jazz harmony. Others might disagree.
Thanks! Just the motivation I needed to give this topic a more earnest shot. I've heard of classical counterpoint before but never looked to deep into it because it didn't seem to be what I was after with a cursory glance.
Real simple example using a song I love would be Porcupine Tree's "Collapse The Light Into Earth." It's a good example because it's 5 chords on loop for the whole song, but everything else really changes the way we experience those chords, and that's what counterpoint is all about.
It's first about music as "voices" and kinda separating out the instruments to just what notes they're bringing to the table. Then it's about stacking those notes together and taking a look at what "chord" exists at any given moment.
Then, if we get nerdy, we start talking about how the individual voice shifts and choices in how to voice lead to specific chords support the... feelings it makes us feel.
So, Collapse the light into earth is 5 chords on loop. They're all triads. Nothing fancy. Bm, Cmaj, Gmaj, Dsus4, Dmaj.
Since I don't have sheet music or a pen and paper, I'll note that, for example, the chorus, Steven Wilson sings "Co - lapse, the light, into earth" The "Co" is a B and lasts the whole measure with the Bm, Nothing strange here. Good support of the well established idea. But then bar 2 of the chorus, "-Lapse" is a D. Over our Cmaj. Making it kind of a Cmaj9, but it's kinda wedged in there with the chord instead of floating over it, so it's kinda spicy. It's no regular C major anymore. And in this way, the Chorus and the Intro are distinct, despite so much remaining the same. Our analyzing the Cmaj as a Cmaj9 is a piece of contrapuntal analysis.
If you listen carefully, you'll notice the vocal melody does this not just in the chorus, but all over the song. And... not just the vocal melody, but the bass... and the synths that come in the later half... The song is a very, very rich example of counterpoint, mostly because you can see how bastardized the original 5 chords look at any point in the middle, despite them being really actually at their core not very far off. Just, embellished. It's fun.
As mentioned by foggy, jazz harmony (which I frankly suck at) or counterpoint are both the things which will give a formal understanding of this sort of thing.
That said I picked up a lot of it more from playing regularly with people who are much better than me at music. In the end if you immerse yourself in music that uses these ideas more regularly you start encountering strange chord notations and seeing patterns in why they are as they are. Finally it isn't really a prescriptive thing, there will always be many ways to write the same chord, and it will usually be much of a muchness what is written vs what you actually play.
In the case above I'd probably always write it as a D because for someone trying to learn it quickly they'll know what a D is more instinctively than a weird augmented minor.