Those who play right are right-players, not playwrights. Well, since they write plays, they should be "play-writes," wright right? Rong Wrong. Remember that a play writer in Old English was called a "play worker" and "wright" is from an old form of "work" (wrought iron, etc.)
A playwright, also known as a dramatist, is one that writes dramatic literature or drama. These works may be written specifically to be performed by actors or they may be closet dramas or literary works written using dramatic forms but not meant for performance.