Play
Play has its own mode of being which we will call a game. This being exists only in the world and is not aware of its own being – it does not have Dasein character. It does however require participants with Dasein character to give the game importance and make it matter.
Players
The players are the participants in the game. In order for them to be players then they must take the game seriously. All players are played by the game and lost to the game while they are playing it. They are concerned only with being-in-the-world, with the they only as it exists within the world of the game. Their 'authentic being' is lost to them. The game can not be disclosed to the world without the players being predisposed to being players.
Playing
Playing is giving oneself to the game and the play - in as much as the game has to be taken seriously one is played by the game. It is essential to be part of the they and therefore it is unlikely for Dasein's authentic being to be disclosed. It is useful to note that this being lost in the they is neither inherently negative nor positive.
Being Playful (moving away from the direct influence of Gadamer)
Being playful allows one to be able to play with what is serious and the games that matter within one's Being-in-the-world. Being playful can make one feel as though one is outside the game or games, it also can transform what is serious into a game. In the constructing a new game, which may be more a complex or a simplified version of it constituents, the pitfalls of other games are created. This game is still part of Being-in-the-world. Being playful is Dasein's conscious construction, and thus is more likely to phenomenologically disclose aspects of 'authentic being'.
Playfulness
Playfulness as a mode of state-of-mind is of Dasein and cannot apprehend itself without losing authenticity. Playfulness envisions the possibilities of our 'authentic being' to be realised within Being-in-the-world, and the possibilities of Being-in-the-world to inform our 'authentic being'. If we accept this then it can be said that Playfulness is a conduit and phenomenologically is of 'authentic being'.
Paraphrasing Heidegger (1926, p180):
That which playfulness plays is that very entity which is played - Dasein.