New Record Coin Price? Are Branch Mint Proofs Real?

The problem is without documentation that they actually attempted to create a special coin all you have to go on is the look of the coin itself. And with branch mint proofs you have the additional problem that they will NOT have the same look as the Philadelphia proofs did. Why? Because the branch mints did not have the high tonnage medal press that Philadelphia used to strike proofs. All they had were the regular coining presses. So they might polish a die, they might polish the planchet, but they really couldn't strike the coin with much more pressure than a normal coinage strike. Then add the fact that new dies were polished to remove oxidation from the hardening process and you could have a lot of trouble telling a branch mint proof for a coin struck with an overzealously polished new die. This is why they are controversial.

They could change their mind, it would just be expensive. It's happened before.

Then there must be an exact explanation of what a proof is. You can't just say "struck with proof dies," there must be an exact definition of what a proof die is. Harder metal? Engraved deeper? Polished more? There also must be an exact definition of a proof strike. Must it be 300 tons? 400? 500? Is it struck twice? Three times? Is a coin proof just because whoever minted it says it is? No?

If these are questions that can't be answered then it cannot be a binary question of fact since it would always be open to interpretation. This is the issue with this coin. NGC says it's a proof, Would PCGS? Would ANACS?

You are looking at it like NGC looks at it. Which is fine. They are unquestionably qualified experts.
Philosophically, and as a matter of transparent accuracy,
I don't think what the coin looks like should be the determinative factor. Besides, there are so many different kinds of proof (matte proof, sandblast proof, deep cameo proof, I'm sure there are others) that we would need several inconsistent definitions to cover them all.
My opinion, and you and NGC are entitled to yours, is that before S.F. started making proofs in 1968, proofs only came from the Philadelphia Mint. No other mint was authorized, had the equipment, or formally or otherwise (1) designated coins as proof; or (2) kept records of issuing proofs. Anything else being passed off as a proof either isn't really a proof or is a knockoff. Maybe it has the "look," even more so than being "proof-like," but the counterfeit designer handbags that are for sale in countries with loose intellectual property law enforcement have the "look," too. I don't see "PF" as a grade. PF-65 is a grade based on an opinion of quality. PF, standing alone is a statement of fact. I believe PCGS sees it more or less the same way, as my understanding is that PCGS doesn't recognize branch mint proofs.
If someone painted an exact copy of the Mona Lisa, indistinguishable from the true Mona Lisa, would it be the Mona Lisa? Not to me, even though it "quacks like a duck."
We know zero about the history of the coin in question, or how in came into existence. We don't even know if it was intended to be a proof, be special or commemorate anything. Giving it the same designation as a true Philadelphia proof seems incorrect, although I can't call it misleading because anyone shelling out the bucks to buy one of these very rare birds certainly knows exactly what they are getting.
It's probably a matter of definitional semantics. If these "special" coins were given a different designation than Philadelphia proofs, "special proof," "branch mint proof," or whatever, it would seem much more accurate and appropriate to me.
Does anyone doubt that officially designated and recorded Philadelphia proofs are really proofs? You ask: "Is a coin proof just because whoever minted it says it was? No?" I would answer your question "yes" with respect to Philadelphia coins and "no" with respect to others.
I guess, to sum up, that NCG appears to view whether a coin is proof as a grading opinion. I see it as a question of fact based on the legal authority and intentions of the coin's creators.

I think we come from different experiential backgrounds. I encountered very few rigid definitions in my lline of work. And factual determinations are often based on opinions.
I do see fact questions that may or may not be answerable.
Fact question: is it intended to be a proof? Intention or state of mind is always a question of fact, but it is a fact. This is not a whim of the maker issue. If the maker did not intend to make a proof and therefore was not trying to make a proof, how can it be called a proof? If it is, isn't it just a debatable matter of grading opinion? I don't know about privately commissoning coins from the U.S. mint, but if it comes from a private mint it isn't a coin and who knows what a private mint would call proof. Do TPGs grade such things?
Fact question: is there any authority, from a person with the right to make the decision, to make the proof at a branch mint only intended to make circulation strikes? This is a fact provable by documentation or personal memories.
It has been so long that evidence relevant to these questions, if it ever existed, may no longer be available. Absent mint records, the best place to research this may be the New Orleans Times-Picayune. In those slower news days of 1891, an occasion important enough to justify creation of a single, atypical branch mint proof might get picked up by the local paper, although the story might not have mentioned the coin.
Obviously, there are no living first hand witnesses. There may or may not be a discoverable oral tradition. Or maybe someone saw something in a document that once existed but no longer does. But it seems the most likely situation is that all that can be found is the circumstantial evidence of the coin itself.
So... does that prove enough to give a common date coin a designation that increases its value by six or even seven figures? Conder101 is correct that the issue is controversial. NGS clearly believes the coin itself is sufficient evidence. I understand that PCGS has come down on the other side of the question, but maybe I am wrong and PCGS, rather than announcing a definitive decision, just has not yet found a coin to certify as a branch mint proof.
I continue to believe that merely looking at a coin with no other context whatsoever can at best be an educated guess and could rise to the level of speculation. But that is just how I see it.

Well I was totally out of my mind on that one ...
I thought I had read it somewhere, maybe it was in a very old book. Maybe it was my feeble mind starting to fail me.
Wouldn't have said it if I didn't think I had a basis for it. Kinda embarrassing. More than kinda...
Anyway, with the two main TPGs disagreeing with me, my rather purist concerns are of no importance.