I think it depends on the person. I've met songwriters who start both ways.
It's almost like meeting guitar players that are self-taught vs. theory based players. Or people who like math compared to people who like art.

Some people I've worked with spend a lot of time writing out lyrics or poetry and the music almost comes as an afterthought.
With me, it's the opposite. I have a hard time even listening to lyrics and vocals in a song and concentrate on the backup musicians.

It kind of depends on the song too. Sometimes you come up with a really great hook or saying and end up writing the music around that.
Other times, the music almost makes you want to say certain words because of the way the rhythm sounds.