My assertion: In D&D, the DM is NOT a player.
1. Common Usage.
2. Division of Roles.
3. The Division of Roles Matters in D&D.
All of those three points, however, are largely irrelevant to the meaning in which "the DM is a player" really matters.
If we set aside the rare case where the GM is actually a paid position, all the people involved are gathered to gain entertainment form engaging in play. In that sense, they are all players. And, that is the important sense.
Yes, there's some division of roles, but lots of games have that. A goalie is not a forward - they have different skillsets, different equipment, and do different things in pursuit of play.
In large part due to Gygax's original take on the relationship between GM and player, we have a tendency to think of the GM as a referee. But, to be honest, Gygax was still locked in the wargame concept, in which sometimes there was, in fact, a referee between two sides of players. But, that's not what actually happens in D&D.
The GM actually makes decisions for units in the play space themselves. THEY ARE ALSO PLAYING. Ergo, they are a player, and we should keep that in consideration more frequently than we often do.