The term "Sound Designer" is used to mean vastly different things by many arms of the arts / entertainment industries. It can get a bit confusing for those, like myself, who work across the boundaries of these divisions.
In Film/TV, Tim's description of Walter Murch and his associates wanting it to mean something like "Sound Director" is correct, and to my mind the ideal usage - though nowadays common usage is often much narrower - the person who creates sounds to match images of fantasy / scifi / animated subjects that can't just be recorded in real life.
In Live Theatre (specifically "Straight Plays" - ie NOT Musicals) the Sound Designer is more often than not, also the Composer when Music is being created to order, as well as being responsible for spot Sound Effects and Ambiences. Whether such a person is credited as Composer or Sound Designer is often a matter of personal preference (or that of the Theatre Company). That same person may or may not involve themselves in the specification of the physical sound system (Speaker positions, processing, playback equipment).
In the Musical Theatre side of the Live Theatre business, the primary role of the Sound Designer is in fact the specification of the reinforcement system, and 'tuning' it to the venue (usually including backstage Comms and CCTV systems). Then they are the person responsible for steering the artistic approach the show's Sound Operators take to the actual mixing of the show. On top of that, they create & plot all required Sound Effects and Ambience. They are never the Composer!
In both forms of Live Theatre, the term was adopted (in the early '70s at around the same time as Murch et al adopted it in Film) - allegedly by Abe Jacob on Broadway, specifically to gain a 'seat at the table' on an equal footing to the other creative team members (Set Designer, Lighting Designer, Costume Designer) who create the show one level below the Director. Before that, Sound was seen as a technical task only - usually an annoying sub-responsibility of the Electrical department.
In Games - the Sound Design role is narrowing to a similar 'creating specific sounds to image' meaning as currently used in Film/TV - with a new role of "Technical Sound Design" being the person(s) who look after implementing those sound elements or 'assets' into the programming framework that runs the game and triggers the sounds when appropriate. the Walter Murch" version of sound design in the Games world would be the "Audio Director" or some similar title - many of whom do 'sound design' work on their projects as well. Except in possibly the smallest teams, the Composer is a separate role, though usually under the direction of the Audio Director (distinctly different to Film/TV - where the Composer answers directly to the film's Director)
In Electronic Music, a Sound Designer is quite specifically the person who either (1) Creates sound presets for Hardware Synths or Virtual Instruments - often for the manufacturer of those instruments, or (2) creates sound resources to be used on such instruments. 'Sound Design' is also used as a description of what a Composer, Musician or Producer is doing whenever he/she 'tweaks' a preset, or ignores the presets altogether and creates sounds using the instrument's controls, their own samples etc.