If someone was staying off-site, their 2-year old turning 3 would be free the whole trip right? Do you need a (single) Disney hotel reservation to leverage the "free" 3rd birthday trip?
Not trying to stir the pot, and it sounds like Disney considers it two stays, but wouldn't this be a disincentive for people to potentially upgrade resorts mid-trip? (But if you are at a Disney resort and register her as 2 at the first resort and she is 3 when you check into the second it would be fairly obvious that she was now a "kid" and should have a ticket)
Our DD will be a recent 2-year old when we go, so our split stay isn't affected by the age issue this trip, but it it was an issue for a future trip, I would make sure to just pick one resort and stay there or stay offsite to avoid the issue. Or you could just buy her a ticket....