Starts with purely being a definition by PL and UEFA - must have been at a club for at least 3 years before the age of 21, and Rose I think joined aged 17 and spent his first season playing for Spurs u18's.
Kids move round Academies a lot when younger before signing up to a scholarship at a club - even Kane spent a season at Woolwich and another at Watford before joining Spurs, and probably would have trialled at a few more. So whilst some kids sign on at some football clubs academies at a young age (eg 8), many do not, which is why the definition came in.
However as
White Van Jan
says the Rose we have today is not the Rose who arrived from Leeds. He arrived as a LW, played in Spurs u18's as a CM, before being converted to a LB (I think whilst on loan at Bristol City) and became established as a decent LB whilst on loan at Championship Sunderland.
So Rose like many kids, are really the product of multiple clubs, in this case supervised on his journey by Spurs from the age of 17 after Leeds almost went bust.
Worth recalling as well that Rose was one of 3 'good prospects' leaving Leeds at about the same time - the other 2 went to Chelsea and (I think) are now playing non league football !