How can you tell if a bird is a hunter of other animals?

First point is that every bird I can think of hunts animals, since fish, birds, reptiles, insects and mammals are all animals and I don't know of any leaf eating birds. So as such the question you are asking needs to be rephrased a touch, but I think you are trying to specify larger prey i.e not worms or insects. The first indicator of this is size: obviously to lift something as large as a rabbit, the bird itself has to be fairly large, but since migratory birds are also unusually large this isn't the only factor. Beak shape is another indicator, but this too isn't without its problems when one shape of beak is adapted for fish, another for mammals and so on. The key indicator is the claws. Birds don't swoop down and catch prey in their beaks, they grab it in their talons. Birds which truly hunter their prey have an additional claw in each leg similar to that of a dog, fixed into the leg unable to be moved but still useful for pinning the prey against.