Separating the two cues wait and stay is not an absolute necessity, of course. Many will just have one standard stay cue to mean “remain where you are and in that position until I release you or tell you to do something else.” This is an adequate way of doing things, but separating the two concepts will usually make each much stronger.
After a quick display of the dogs playing on cue, this video shows a solid stay, then demonstrates wait as a cue to remain at the current location, in the current position, and with attention on the handler. If the dog is standing, then it is all right for them to sit or lie down, but if already prone or semi-prone, then they should remain in that position. Further, wait may be broken or modified by a recall or position- change (such as from sit to lie down) cue from a distance. Wait is different from stay in that stay will only ever be released or transitioned from within touching distance. When on stay, the dog’s attention may wander from the handler, as the dog may learn through repetition that they will be staying put until the handler is within touching distance. When on wait, the dog’s attention should remain on the handler, as a cue or release will be coming soon.
A good way to teach these is to begin with wait. Put the dog in a sit or down position, then turn your hand parallel to the ground with the forefinger and middle finger extended and give the cue wait. When first teaching this, over-enunciate, and lengthen the vowel sound, “Waaaaaiiiiiiiit.” As in all training, speak clearly but gently and with a positive tone. The first threshold is just one second and one step backward, then two seconds and two steps, then five, ten, etc. It is helpful to randomly alternate between returning to the dog and clicking and treating or calling the dog to the handler. Once the dog can maintain the wait steadily for a few seconds while the handler is a several feet away, increase the challenge by walking around the dog halfway, then returning to the dog’s front. Once this can be done reliably, walk a full circle around the dog. Gradually extend the distance between the handler and the dog, alternating between calling the dog and returning to the dog.
Learn stay in much the same way, but only use stay for longer periods (like a minute or more, and building up to 30+ minutes), only release or transition the dog from within touching distance. The hand signal is an open hand, palm forward, like a traffic-cop’s “STOP” signal.
Examples: cue the dog to wait when preparing to open a car door, or when feeding, or any time the dog may be called to the handler. Cue the dog stay only when the handler will be returning to the dog, and for longer periods, or when the handler will be out of sight from the dog.
Preceding the stay and wait demonstrations, this video shows the handler cueing the dogs to play. We didn’t really teach this in a formal way, it is just a gesture and a word that we said again and again when the dogs were encouraged to run around and be crazy – now they know the gesture and the word as a signal to start frolicking. This is a good way to help them burn off some energy, as many dogs won’t really exercise unless prompted to (which is why leaving a dog in a backyard, even a big yard, is not an adequate way to get the dog exercise).