I get that you’re arguing that, but in your argument you literally stated that the DM is not a player in your games because of those play preferences you listed, which implies that you are assigning baggage to the term “player.”
Not "baggage", just context, which was my initial point. I also stated that "...the GM is very much not a
Player..." (in my most preferred games) which is quite different that what you are saying I literally said, even if it is quite similar.
No, to me it’s important that the DM be acknowledged as a player because I believe the game is supposed to be fun for the DM too, and losing sight of the fact that they are a player in a game de-emphasizes the importance of the DM’s enjoyment of the game. DMing isn’t a service one performs on the player’s behalf, it’s a leisure activity one engages in for their own enjoyment.
Ok, I can understand the point of this much better than the ipso facto argument. But I would make a few points;
Having fun isn't predicated on being a "player", at all. You can have fun being a GM in a game, watching a game, or freeform play outside of a game, heck you can have fun working.
GMing doesn't have to be about having fun, in a particular moment or even overall. It can be about delving into serious issues, it can be therapy, it can paid labour.
Maybe I should just stop there and get to the larger point; "The GM should have fun because they are a player" just isn't very compelling to me in this context. "Fun" and "player" just aren't that inextricably linked
Who exactly is saying that the DM is a Player? All I've see is people saying that the DM is a player (which is a different beast).
People, probably, who knows. Earlier in this post you will see where the distinction was lost, it's not hard even when the subject of discussion. Seriously though, what I think is important is
why somebody thinks it's useful to say the GM is a "player", even if they are not a "Player", or the other way around. My point is that the ipso facto definition isn't useful here.