That’s where the issue lies,your approach places everything into what I would call the second category: metagame agency Meta Agency. In doing so, it ends up being more limited than mine.
The reason is that RPGs aren’t games in the conventional sense. Game theory has only narrow application to RPGs. What makes tabletop roleplaying distinct is its character-centric play loop: the referee describes the situation, the players describe what their characters do, the referee adjudicates, and then describes the resulting situation. This cycle, pioneered by Dave Arneson in his Blackmoor campaign, is the core of the experience.
Because of this character-centric loop, there's an entire layer of agency in TTRPGs that isn’t present in most other games,specifically, what players can do solely as their characters.
You can see this difference at the very beginning of the hobby. Blackmoor was the first tabletop roleplaying campaign. Dave Arneson demoed it for Gary Gygax, which inspired the creation of OD&D. At the same time, Dave Megarry, inspired by the same Blackmoor dungeons, created Dungeon!, a board game. Both were born from the same source, but one clearly remained a board game while the other created an entirely new type of experience.
Further, it's possible to run a tabletop RPG campaign where players never interact with the rules at all. If the referee is skilled and has solid preparation, they can adjudicate everything using only their notes. It’s not easy,it demands strong communication and a high level of trust at the table, but it can be done. I’ve done it twice in 40 years of refereeing, both times as one-shots. Each felt like a typical TTRPG session using a rules-light system.
I understand you may have objections or questions about this. But ultimately, I think it’s a mistake to view RPGs solely as games. They are their own category, and “game” is only one part of what they are.