I'm just going to have one small say in the matter of whether this is a game or not...
This is the game "Snake", of which there are endless variations:
Most people will agree that this is a game. You tell the snake which of the four ways to go, but it's always moving in one direction, and you try to pick up small dots that make the snake longer.
Now let's compare that to Dear Esther...
You play as a man on an island. You tell the man which way to go, and you can make him stop and take in the scenery on the way if you so wish, while you try to trigger voiceovers that adds to your understanding of the story.
Take the Snake game, switch dots(or apples, eggs, whatever) with voiceovers, switch the increasing snake body with increasing understanding of the story and game world, add more control, and better graphics and audio, and you've got yourself Dear Esther.
Break it down, and you've essentially got the same game in both. Move, trigger, repeat until end. In other words, Dear Esther is as much of a game as Snake is. It doesn't have a great amount of interaction, it's fairly simple in its gameplay, but even so, it is a game.
Cheers!
