No, an engine is only the code that is shared between games. For instance "urban terror" by frozen sands is a closed source game on top of a (now) open source engine (the Quake3 engine). Quake3 is a (now) open source game (with closed assets) on top of its own open source engine. Quake and it's derivatives are a particularly nice example of the distinction between a game and its engine because the engine includes an interpreter for the game code, leaving a very clear boundary. Meanwhile many games end up intermingling their engine and game code (e.g. see the Eve Online devs write about this - another closed source game on top of a now open source engine).

BAR is an open source game. All the game source is open.