31st July 1991 (to be precise).
The Stone lived in Gringotts until such time as Flamel grew worried that someone might try and steal it. At that point he asked Dumbledore to look after the Stone.
"See?" said Hermione, when Harry and Ron had finished. "The dog must be guarding Flamel's Philosopher's Stone! I bet he asked Dumbledore to keep it safe for him, because they're friends and he knew someone was after it. That's why he wanted the Stone moved out of Gringotts!"
(Philosopher's Stone, Chapter 13, Nicolas Flamel).
Dumbledore subsequently delegated the task of collecting the Stone to Hagrid.
"An' I've also got a letter here from Professor Dumbledore," said Hagrid importantly, throwing out his chest. "It's about the You-Know-What in vault seven hundred and thirteen."
The goblin read the letter carefully.
"Very well," he said...
(Philosopher's Stone, Chapter 5, Diagon Alley).
Hagrid picked up the Stone and delivered it to Dumbledore. He did this on Harry's 11th birthday and was only just in time as Quirrell looted the vault later that day.
GRINGOTTS BREAK-IN LATEST
Investigations continue into the break-in at Gringotts on 31 July, widely believed to be the work of Dark wizards or witches unknown.
Gringotts' goblins today insisted that nothing had been taken. The vault that was searched had in fact been emptied the same day.
(Philosopher's Stone, Chapter 8, The Potions Master).
Concern about an imminent break-in was the only thing that prompted Flamel to look for a new place to store the Stone. Up until that point he had no need to give it to Dumbledore or anybody else.
As for who owned vault 713, I'd say unknown but probably Flamel's. There's no canon evidence on this that I'm aware of but I've always assumed that it was Flamel's vault. It would've been relatively easy for him to access the Stone and therefore the Elixer of Life if it lived inside his own vault. It would've created an unnecessary hurdle for him to keep the Stone in somebody else's vault. He would've had to have gone through the hassle of borrowing their key and returning it to them etc. - much easier to keep it in his own vault. It makes sense to conclude that vault 713 was Flamel's vault and that he only moved the Stone from there when he knew that somebody was after it.