I think we differ there too.
Exploration is the act of characters interacting with and thereby learning about the world. When organizing the events of a night's gaming session into a story, much of the exploration pillar is emitted from for brevity, as is the case when reading the Lord of the Rings or every other fantasy novel. That is because stories are rarely about exploration; instead, they are about the conflict and resolution of completing priorities. I therefore differentiate exploration from story.
I think this is too hasty. If you remove the setting and locations from Lord of the Rings, does the story remain unchanged? I don't think so. Setting is a character all it's own in stories, and you can't just separate the stage from the play -- it has it's own very important role in the story. I think you might be meaning plot when you say story, which I think might be more defensible, but I'm not quite willing to concede there without a lot more thinking on the matter.
Regardless, exploration is a key part of RPG play. Even in my Blades in the Dark game, which features a very confined setting (or appears to) and where play is laser focused on the PCs doing their thing, exploration of that setting happens all the time. Finding out things exist in the setting is exploration, discovering that this building has a secret basement full of old things is exploration, etc. These definitely impact the story that's emerging from play. In 5e, that exploration is even more explicit and critical to the unfolding of the story of play.
As far as the Social pillar is concerned, I agree that characters are used - both by the players and the DM. Nevertheless, I still believe character creation and character sheet maintenance happens before or between periods of gameplay. I see leveling up (character sheet maintenance) as a consequence of gameplay, not part of the gameplay itself. Football is the same, points are awarded for skillful gameplay and have an affect on the gameplay, but updating the scoreboard isn't really part of playing the game.
I disagree. Even if character creation/levelling doesn't occur at the table, it's very much part or playing the game. The game has clear rules for when and how leveling occurs -- to do it I have to refer to the rulebook! In this, leveling your character has nothing in common with choosing to keep score in football on the electronic scoreboard. Keeping score is part of the rules of football, though, so you're confusing a choice of method with doing the required play. Your analogy to football scores is more apt for saying that choosing between paper and electronic character sheets isn't part of the rules. Keeping and updating the character is part of the rules, but the specific manner of doing so isn't.
Instead, I would define the social pillar as negotiation, whether between player characters or between player characters and NPCs. Just as the combat pillar resolves conflict by force, the social pillar resolves conflict through trade-offs.
I think this is fair, although I think that trade-offs occur no matter what. The nature of the trade-offs vary between resolution methods but they always exist, even if just in opportunity costs. There are trade-offs in combat and in exploration, so while I understand the gist of your statement here, I think it's framed in a way that's too narrow and unappreciative of other trade-off occurring in other pillars.
To address the topic of pillars, I think exploration often gets ignored because it's not as explicit as the other two, or, rather, that we don't notice it when it happens. Here's an example -- let's say that we want to get the pie from NPC Orc Bob. How does this look using various pillar approaches?
Combat: pretty straightforward. Bob has pie, we want pie, we have sharp metal. Insert sharp metal into Bob repeatedly, get pie. Tongue in cheek, but it's pretty easy to recognize the combat pillar because it has violence as a result.
Social: Again, straightforward. Bob has pie, we want pie, we have pretty words and maybe something to trade for pie. Again, if two characters are interacting absent violence, this is usually going to be a social encounter. We easily recognize these.
Exploration: here's were we miss that the following is exploration. Bob has pie, we want pie, we're very sneaky and sneak in and steal pie from Bob. This is exploration pillar -- the challenge isn't to convince Bob or fight Bob for the pie, but to use the environment to aid us in gaining pie from Bob. This is quintessential exploration of the setting -- does it exist in a way that will allow us to get the pie from Bob? If the GM says Bob and the pie are in the middle of a featureless 10' x 10' room, then this is probably not going to work, but that's because we've explored the setting to find this out. If the pie is in Bob's bedroom in his nice townhouse over a shop, and Bob is currently working in his shop, so we can climb to the alley window, sneak past Bob's lazy basset hound Bob (Bob thinks it clever that his dog's name is just Bob's name backwards), pick the lock on the display case, and sneak back out then this is also very much exploring the setting. The former example of the featureless room is like boring combat or forced social encounter outcomes -- not a good example of play in this pillar. But it is still play in the exploration pillar.