Consider this example: A hypothetical game exists which has a very hard difficulty mode that has a permanent death mechanic. If you play on this mode and fall off the first ledge you see, you will see a short cinema of how the world is doomed because you died, then you see a "The End" screen and must start a new game. Any achievements for completing the game are not unlocked by doing this. Does this count as completing the game in a few minutes for a speed run? Is there a difference between reaching a permanent game over screen as opposed to completing the game?

If the game doesn't reward achievements clearly labeled, "Complete the game," does it still count as completing the game?