I don't think there's any actual 'correct' way to do it. It just depends on your personality and whether you're more language oriented than musical.
It's kinda like there's two types of writers. There's the ones who are always writing poems and putting their feelings into words. Then they write music later to fit the lyrics.
Then there's the other type who tend to write most of the music first and then fit lyrics to the music.
A lot of artists have an idea for a melody or a hook, and some ideas for what they want the backup music to sound like. They record the music first, then sing nonsense lyrics over top of it so that whatever is coming to mind first is what comes out. Then they go back over the tape and gradually come up with phrases and words that fit the nonsense sounds they're hearing in their heads.
I've got a couple of bootleg recordings like that. No idea where they came from but one of them is M. Jackson singing Billy Jean. All the music is done and he's just singing stuff like "Billy Jean is not na na na... she's just a...na na la and now is the one... but a furl is gone a gun"
Just anything that fits.
Then over the course of the next few days or weeks you just keep going over it and refining it. Ideally you come up with a lyric that not only makes sense but has a hidden meaning.
A lot of the time you come up with a bunch of abstract words that don't really mean anything, but tend to take the listener on a journey ie, I Am The Walrus by The Beatles... sounds like a lot of the nonsense lyrics were just left in because they sounded good.
Paul McCartney has often said that the original words to "Yesterday" were "Ham and Eggs" or something...