The 7d Mark II, hands down. It was designed, in part, with birders in mind. If that's out of your budget, probably the original 7D, though that is arguable for a couple reasons.
Edit: just noticed you mentioned "lightness" as a plus. If that's a big factor then you have to make compromises, and it turns into a much bigger discussion of various options. THe 7D series is not light, not by a long shot. It's very fast (10 fps), and has a terrific autofocus system. But if it's too big and heavy I'd actually recommend the other end of the spectrum - the new T6i Rebel. Autofocus and speed is going to suffer tremendously, but that doesn't mean you can't capture Birds in Flight (BIF) with it. It's going to be smaller and lighter than any other option, it has the latest sensor technology, and has more pixels than all the others - better for cropping. Yes, you're going to miss some shots that the 7d2 would get, but it's still capable in the right hands.
Other people would argue for full frame cameras. Shooting wildlife is often at dawn/dusk, and you need a fast shutter speed to freeze the action, so the better low light performance is helpful in producing acceptable quality images. This comes at the sacrifice of a wider field of view, and BIF photographers almost always crop. THe only light full frame is the 6D, which has a very simple AF system. It's designed for studio/portraiture/landscape type work, but I know several people that love it for wildlife for the reasons mentioned above. A camera like the 5d3 is going to be far more suited to the task, but again, it's not small and light.
