Are we really arguing about a definition, or are we arguing about a role?
If you're trying to decide how to define player in the context of D&D, it seems like a waste of time as it makes no meaningful change. Whether DM is included in "players", or people should use "players and the DM" when discussing a situation involving everyone at the table is incredibly silly to work about - context of the conversation will tell you.
If this is about the role of the DM, and the duty the DM has to others at the table, then focusing on the language used to describe the DM is only tangentially related.
Either way, this is an odd conversation to carry on for 16 pages.