If there is a gradient from a Budgerigar in a cage in a home to a wild flock of Budgerigars flying around in Australia, I would consider a Budgerigar flying around free here in Canada to be closer to the one in the cage. It’s not part of a viable population (it clearly came out of someone’s house), it can’t breed, and it can’t survive on its own because of the winter. But not too far south of us in the US, there are populations of Monk Parakeets which do make it through the winter. Then a Monk Parakeet in exactly the same situation as the Budgerigar is a bit more complicated.
I feel like for consistency between taxa, if an organism was initially in “captivity” and then “released”, it should remain “not wild” throughout its life, and its offspring all count as wild, but that might not really be practical…
Perhaps ratings or a grey area as suggested here would be the best option?
I would interpret that quote from the help page to mean that individuals in a population descended from reintroduced individuals would count as wild, but it’s unclear about the original reintroduced individuals.