To properly start a game on Hard difficulty, you need to change the difficulty from the Options Menu, and then start a "New Game" (which abandons your current session).
If you choose to set the difficulty level once you exercise control of Cole MacGrath, you already failed the criteria of starting a New Game on Hard. Set it, then start it to satisfy this Trophy condition. (Note: Tempt_Not_the_Blade claims the difficulty must be set as soon as the menu can be brought up at the very beginning of the game.
The player MUST WAIT until credits have finished rolling and 'Congratulations' screen appears to meet requirements. Wait a few seconds and the Trophy will be awarded.