I use "social environment" a bit differently than social game or social system: to mean a setup in which multiple games can be played. You can think of this as a partially-specified social system, if you like. If you're designing an organization, an event, or a venue, you're probably designing an environment, rather than a single game/system.
A "social design" can mean either—a game/system or environment. (The word "institution" can also mean either.)