there is a difference between a song being a song that Christians can associate with and a song that is an actual Christian song

Any song that mentions Jesus (favourably!) is probably A Christian Song, although some of them are pretty heterodox (take the Cherry Tree Carol or Under the Leaves for starters). Beyond that I'm not sure it's a question that's even worth asking. If you're an X-ist and the song works for you and your X-ist friends, it's an X-ist song... for you and your friends. (And of course it's open to the Y-ists down the road to claim the song as theirs; it may work for them, too.) That doesn't make it intrinsically an X-ist or Y-ist song, or intrinsically anything else. All it is intrinsically is a series of sounds.