Move 6 = A loss

My opponent made a blunder on move 6 that cost him the game.

Not really, the move was not good but after 13. a3 the position is about equal so I would not say it cost him the game.

That's why we always improve

Move 6 = An Inconvenience

Be6 is a strange move and gives white a tempo, but I wouldn't consider it a "blunder" :)