When you post a song you made with BIAB, can you call it your song? When you go into the studio and create a song live with other musicians, can you call it your song?
I submit that the only way to say a song is your song is to play all the parts live and record it, including the vocal. Even then it is likely not very original because you based the song on what you learned from other people.
Absolutely, yes. You wrote it, therefore it is your song no matter how, or who records the tracks. It's not a requirement for me to record everything myself to call a song that I wrote, completely, my song. Regardless of what I learned from other folks, my piano teacher as a child, the drum teacher in school, the musicians I have played with, and the music I listen to on the radio, unless I am copying something, if I thought it up on my own, it is fully my creation.