It's interesting how different bands approach this issue. In the case of the Beatles, Lennon and McCartney agreed to share songwriting credits while the band was together, regardless if either one wrote the entire song. The Doors shared credits on everything with everyone in the band, though Jim Morrison wrote most of the lyrics. (A notable exception was Light My Fire, written by their guitarist Robby Krieger).
Not sure what the law is here, but it would seem to me that if Copeland and Summers allowed Sting to keep the rights to the songs, that's a bad move on their part.
Isn't there two different kinds of royalties? Composer (or lyricist) and performance royalties? You would think the band would get something.
Bruford chose not to play at all during the improv, leaving only the bassist, guitarist, and violinist on the entire song. The band thought his decision not to play on what turned out to be a rather delicate acoustic improv was so critical to the success of the piece that they gave him a credit despite him not playing at all.