I once heard an obviously over-educated DJ on an oldies station back-announce Bobbie Gentry’s “Ode to Billie Joe”,
“I’ve been playing songs all morning, but that’s the first ode”
I play in an instrumental band: no singer, just players.
I don’t call our pieces “songs” because there isn’t anyone singing on them. I don’t recall hearing anyone else referring to our music as “songs” either.
I don’t have a particular term for our pieces other than “pieces”, or “bits” or “things” if I’m feeling less formal. I’m sure I’ve called them “tunes” too.
I don’t think of myself as a songwriter, either, since I write music but not words. I call myself a composer without feeling particularly self-conscious. “Composer” (in this context) literally means “someone who writes music” and it doesn’t imply that one is composing classical music in particular. I play with classically-trained people, so I use classical-music vocabulary because it’s a handy way to communicate.
“Take me to your lieder!” 
– which may actually be the answer we’re looking for. A lied does not necessarily have words, just a melody suitable to be played solo or vocalized, as opposed to symphonic or multipart pieces.
Yes, because birds sing "birdsongs". 
But, can you make a guitar sing?
If one can, Eddie VanHalen surely can.
Yeah, regardless of its original meaning, “song” in common parlance long ago came to mean any musical arrangement, I think. And I can live with that.
I agree 100% with all of this. And I find it ironic that you are the only one accepting a non black-and-white position for once.