As you said, everyone does it differently, and that’s because some people are more musical, whereas others are more verbal.
Even though I’ve written a couple of instrumentals, most of my songs (99% of them) started off as poems with some sort of internal rhythm. In many cases the music was added later, and in most of those cases it’s just some simple three-chord melody (very similar to your average Bob Dylan tune).
There was a period in my life when I tried harder to write better music, and then I’d generally start off with a phrase or the first verse, add music to it, write the music+lyrics for the chorus together, then put it all together and add some solo or something.
On some rare occasions I might have a melody in my ear (but it could be from something I’ve heard) and end up adding words to it. Usually those songs are crap though. Or at least the lyrics are crap.