Yes, you have covered the ground and added more dimensions! I am two-dimensional at best--that is, a vAggEyakAra.
Here's an incident from years ago which informed me that I was one:
Someone: iva compose ellAm paNNuvA, theriyumA? She's a composer.
Someone else: No, she's not! A composer has just the words. A vAgEyakAra is both (the same derivation given by you was quoted).
I also want to say that some composers (like Ambujam Krishna) did not tune what you hear as a song now. True, but all the same, did hum it in 'some' rAgA--may be the same one as we hear it in. That dimension in them is not obvious because someone else tuned their songs. I think, just as words come naturally to us humans, music and beat do too. That's why when a song isn't lyrical but is just a string of words, it's difficult to sing it or tune it. My two cents worth.
I think, it's like the ever-evasive thAlam in me. It's there, I know, but it eludes me--and almost always, indicates the tAlam to the singer with no trouble