The same point ... that is the OP. Because the OP was written by an attractive, articulate, indubitably correct person that likely smells like fresh laundry.
ahem
Seriously, though, that was the first section of OP, but you were just doing it in reverse. It's the pedantic point of definitions and usage.
A: DMs and Players are both "Players" because they both "play" the game, and someone who "plays" a game is called a "player."
B: DMs are not Players because there are two distinct groups that are referred to in D&D; "DMs" and "Players", and referring to the DM as a Player (or vice versa) is bizarre and confusing, and no one would ever do it.
Notice what's going on? Yes, if I DMd, I would also say that I "played" D&D in the broadest possible sense- because D&D is a game, and I acted in the game, and therefore I "played" the game. Because we lack that level of fine distinction in our word choice.
OTOH, if I am discussing D&D, I would never confuse "player" and "DM" because those are two completely separate roles within the game. Some historical examples:
Might & Magic states that "Number of Players: At least one referee and from four to fifty players can be handled in any single campaign, but the referee to player ratio should be about 1:20 or thereabout." (50 players!)
Moldvay puts it like this: "When a group plays a D&D game, one person acts as a referee and is known as the Dungeon Master (DM). Others play the roles of fantasy characters and are called the players."
There is a lot of confusion between two similar, but different, terms: player (as in a generic "one who plays" and doesn't like playa haters) and player (the role of player in D&D).
Saying that DMs are players needlessly confused the two terms. IMO.