You could try reading Jerry Coyne and Allen Orr's book "Speciation". Executive summary: there's about 20 different ways that scientists classify species. Fortunately (for us here) there's only 2 or 3 which are used by ornithologists, and BSC is leading the pack.
Probably others in the thread have said something like this, but here's how I see it: at time X there's a group of birds which are freely interbreeding. A million years later, due to various external influences, there's two groups which breed freely within the groups but not between the groups. So speciation has occurred. But during that million years it's been confusing, both for us and for the birds. Hybrids may or may not have been produced, depending on what those external influences were.
But there's no instant during those million years when we can say "Then! That's when it happened!" There's only a long grey area when it happened. This implies that we are in that long grey area for many speciating groups. But we expect the working scientists to provide a yes-or-no answer when it's really not possible to do that, which results in the discussions which you can see in the literature.
(It's actually more complicated than that. But I think this is a reasonably good overview.)