There are two alternatives to this list, located at Talk:List of songwriters/temp and Talk:List of songwriters/temp2. Some refactoring would be good.
This list of songwriters includes names of those who have written the lyrics or music, or both, for successful songs. It is not limited to popular music.
Songwriters who perform their own work -- a relatively recent phenomenon -- are usually called singer-songwriters. (See List of singer-songwriters.) This list only includes those who remain in the background and rarely, if ever, perform(ed) their own songs in public.