First, I think it is important to mention that the Turing Test as is currently accepted is an updated version of Alan Turing's proposed imitation game, so your question is twofold. I don't think Turing makes this distinction that you propose. To Turing the question was "Can machines think?" and he explores this by way of the imitation game.
The new form of the problem can be described in terms of a game which we call the
'imitation game." It is played with three people, a man (A), a woman (B), and an
interrogator (C) who may be of either sex. The interrogator stays in a room apart front the
other two. The object of the game for the interrogator is to determine which of the other
two is the man and which is the woman. He knows them by labels X and Y, and at the
end of the game he says either "X is A and Y is B" or "X is B and Y is A." The
interrogator is allowed to put questions to A and B
It is A's object in the game to try and
cause C to make the wrong identification
The object of the game for the third player (B) is to help the interrogator.
This is Turing's outline of the imitation game (shortened) between a man and a woman. The only difference between this game and the one we care about is that we replace party A with a machine and the only requirement on B is that they are human.
So yes, in Turing's imitation game the machine is privy to the information that it is not actually human. Whether the machine uses this information is up to the machine.
To talk about your statement on pretending to be a duck versus believing you are a duck lets examine the base imitation game with ducks and frogs. Let's say player A is a frog (either pretending to be a duck or truly believing that it is a duck) and player B is a duck. Suppose for this example, the frogs and ducks are able to speak English and follow the rules of the game. The interrogator may ask "what is the color of your beak?" The duck will respond yellow, assuming its beak is yellow (I am no bird expert), and then our smart frog pretending to be a duck will look up the color of the beaks of ducks and respond yellow as well.
However, our frog that believes he is a duck will think "I am a duck so to discern the answer I must only look at my beak." And thus, our frog will say "I do not have a beak, but my mouth is green." This will oust the frog.