That is correct, the title is currently misleading (arguably the title of every paper I ever wrote was misleading just after I wrote the title, I guess). If you are interested in seeing more details of the proof I'll be following, they are here https://web.stanford.edu/~dkim04/automorphy-lifting/ . This is a Stanford course Taylor gave this year on a "2025 proof of FLT".

It's possible we never found the one he had, but it's pretty unlikely given how many brilliant people have beaten their head against this. "Wrong or joking" is much more likely.

The former.

We can't be 100% certain that Fermat didn't have a proof, but it's very unlikely (someone else would almost surely have found it by now).

Its a "dog ate my homework" situation

Fermat lived for nearly three decades after writing that note about the marvelous proof. It's not as if he never got a chance to write it down -- including proving the specific case of n=4. (So it sure wasn't his "last theorem".)

There are many invalid proofs of the theorem. It is practically certain that Fermat had one of those in mind when he scrawled his note. He realized that and abandoned it, never mentioning it again (or correcting the note he scrawled in the margin).