Shouldn't matter where they stay, though the policy DOES seem to center around the "start" of your vacation based on check-in.
Honestly, they aren't very militant about checking ages for kids like that. They seem to have a general trust in their customers. They probably want a good customer experience over the additional revenue they could get by aggressively checking ages and forcing people to prove it.
That being said, I am sure there are a lot of families that take advantage of this policy on both ends of the children age range.
My son turned 3 the day he went to MK, but our "trip" (on-site) started the day before. I purposely planned the trip around those dates to save money on his ticket AND just to be there on his birthday.
We were never questioned on his age .. we were waived right through every time. I didn't have any proof of his age with me, as I was confident in the policy (based on feedback on this forum).
He had a Magic Band just to fit in (they sent us one), but there was no tickets or Fast Passes associated with it. The only time we had an issue was a buffet meal at Trail's End where they just charged my son a kid's price (this was 2 days after his birthday), but when I questioned it ("You don't age at Disney World") they just immediately waived it (as all kids under 3 eat free at buffet meals).
I would go without buying a ticket and unless your grandchild is big for his/her age, I doubt you will even be asked their age.
