Traditionally, songwriting refers to the writing the core structure of the song. This could be done vocally, but traditionally is done on an instrument. As I said before with STP the process almost universally begins with one of the DeLeos writing the base structure on guitar, so whoever does that gets credit as a songwriter.
I'm sure at the end of the day all four band members, as well as Brendan, helped shape some of the songs. But a lot of the stuff they wrote came out as a complete idea on acoustic guitar, before being introduced to the rest of the band.
But regardless of who wrote what or where each idea came from, the thing about STP back in the day was that it was greater than the sum of its parts. There was more to the band than just the Deleos' tone, playing & songwriting; Scott's iconic voice & sense of melody & genius lyricism; and Kretz powerful, behind the beat drumming. All of those things together somehow added up to something completely amazing and intangible, that was transcendent of what all those things should add up to.
The only reason I ever supported STP with Chester was because Scott had made the band the sum of its parts -- or worse, less than the sum of its parts. The reunion wasn't STP, it was STP with Scott Weiland. Suddenly the band was all about how Scott's vocals and the band's playing just didn't match up, both in studio and live. There was a staggering lack of the chemistry they had once had.
But with Chester, they were a complete unit again. Once again they were all playing on a level that was better than they could each achieve individually. Chester had the chemistry that made the band at least something of what it had been, he had a passion for the music that Scott-- and quite frankly the rest of the band-- lacked after the reunion. He had the vocal range to sing almost any STP song, and a knowledge of those songs to sing them at the drop of a hat. His one lacking was writing lyrics and melodies... But as I said, if the rest of the band stepped up and wrote melodies and lyrics, with Chester singing, then maybe they could have released the closest thing to an actual STP album we could have ever gotten after 2008.