Per the rulebooks, the DM is considered to be a player, albeit a specialized one. Categorically speaking, however, a specialized type of player is nonetheless a player.
I disagree. The OP cited the PHB provision in the OP, and given we should assume the OP probably reads the sections that they cite (because it is a courteous thing to do, and because the OP is likely super handsome and smart and stuff), the OP was likely aware of it. The DMG is similar- but allow me to illustrate the issue:
"The DM creates a world for the other players to explore, and also creates and runs adventures that drive the story. ... As the player who creates the game world and the adventures that take place within it, the DM is a natural fit to take on the referee role."
Compare
"A bunch of players get together to play soccer. If there are sufficient numbers of them, one of the players might take on the referee role."
See what I did there? Is the player shifting roles within the game, or taking on a new title? What is the context of this shift? I mean, we might as well get into a debate about "natural language" in D&D rules, or maybe start discussing Wittgenstein or Gusdorf (if I say "red" does that mean the same thing to you that it does to me?).
But at a less abstract level, this goes to the natural confusion we have given that there are two different terms with two very different meanings that both employ the same word.
To channel myself in fifth grade, answering a vocabular quiz, a player is a a person who plays at something. It doesn't have to be a game. You can have a soccer player, a guitar player, a monopoly player, even an actor can be referred to as a player (and a play is a play). The usage is broad enough that while it has roots in entertainment, it can be used generally. "Watch out for startup company; it will be a major player in anvils soon enough."*
On the other hand, D&D uses the term player in a specific sense, to mean the group of individuals who control PC (player character); in effect, player in this sense is closer to a synonym for "adventurer." Notice the the OP has two sections, titled "Division of Roles" and "The Division of Roles Matters in D&D".
This leads to the final sentence in the OP:
Again, there are other TTRPGs that are built in a different way, but when it comes to D&D, there is a distinction between players and the DM that is useful to maintain, both as a matter of language and in terms of the roles that they play at the table.
That leads to the entire reason for the OP, and the debate in this thread. It's not about quibbling over semantics. It's about the unaddressed claims that come when someone says, "The DM is a player like any other player." There is a lot of normative issues that are behind that simple statement.
Most of them having to do with divisions of authority, and issues of world building, and how people view the three steps of "How To Play" in the PHB, and so on.
So now, contextually and categorically, within the context of D&D, I would say that the DM is not a player. For the reasons listed in the OP and expounded upon the followup posts. But in saying that, what I am saying is that in D&D, there is a distinction between the DM (referee) and the players (adventurers) that is useful to maintain, both as a matter of language and in terms of the roles that they play at the table.
*A DM who is strumming a guitar at the table is a "player," but I don't think that's a very fun or rewarding conversation to have.
