Check out Greyhound Downs 101 and Greyhound Downs 102 and Greyhound Downs 103 if you have not already.
Once my greyhound understands my release word, easily responds to my down cue, and yields to a little collar pressure, I start working on proofing my stay.
I once saw an extreme obedience reality show on Animal Planet. One of the tests was to place your dog into a sit stay and then an elephant was brought in very close. Obviously, none of these handlers had prior opportunities to practice stays with an elephant as a distraction. However, the successful dogs had been proofed enough that they trusted that they should remain seated and ignore the elephant in the room. This is why proofing is so important. Proofing convinces the dog to start generalizing. Initially, a dog will use anything as an excuse to get up. But as your training progresses, there is a point when the dog will start to choose to remain in the stay despite the new distraction. I would bet that Katie could stay with an elephant in the room.
Here is a video clip of some basic proofing to start you off.
I honestly do not tell my dogs to stay until they are really offering good down stays.