I've noticed several times that a printed chord like a C2, or G2 leaves me wondering why it was suggested. Maybe it's the way my brain processes things, but it occurs to me that what's really being called for is a C9, or G9, and the reason is because the melody goes up to that note, and this way the guitar or keyboard in question can support that vocal interval.
I also know that it's not always convenient mid song to stretch a ninth on a keyboard, but it is if you're playing the first inversion of the chord, and you can also drop the root. So G9 becomes B-D-A in that order. I realize that there would probably be another way of notating it, but nothing that anchors my brain as well in terms of where the melody is going. Especially if I am the one singing the lead while I am playing.
Sorry if this is too Theory 101 for some of you, but I'm hoping it might help someone who comes across those '2' chords to see how they integrate with vocals.
Edit (one day later) -- Well, I've been schooled! After all these decades, I did not realize that C9 is shorthand for what I might have referred to as C79. Or Gm/C. As commented below, I was trying to "describe upper structures" on the keyboard especially; the ideal of supporting the vocalists with that extra note which breaks out of the original octave. How about G5/C? (Lots of parallel fifths there for the purists.) Working with non-professionals in our worship teams, I find that the vocal support needed simply isn't suggested by playing C2. The D-note is there, but sometimes it can feel like you're singing against the C major chord, whereas if you have the lead note actually being played in relation to where the melody has just been, it strengthens the note's (and thereby the lyric's) impact.
Much of our notation grows out of jazz, and I can see where in that context, it might be more useful for the specific designation of the 9th interval to include a 7th (major or minor) but after all this happened yesterday, I asked someone to name the notes in a C9 chord, and they said the same as I had thought and were shocked that the 7th would be introduced.
Sorry. I was just trying to help. I won't delete the post however, maybe someone can learn from this after they untangle it.