I suppose I need to answer with 2 sentences because "beating the game" and "completing the game" are two entirely different matters for me.
I would consider "beating the game" as: Beat the Elite 4 and Champion. (This is my "Completed" game definition) Adding additional events after are merely things to keep you playing the game.
"Completing the game" implies that you've done EVERYTHING in it, as in: Nat'l Pokedex completed, Battle Facilities (i.e. Battle Frontier, Black Tower/White Treehollow, etc.) "completed" (all prints, all floors cleared, etc.), post-game Legendaries caught, post-game Trainers defeated, etc. (This is more of beating the game in a completionist's viewpoint, imo)