People use "positive training" when they mean "reward based training".
For prey animals - one of their fav rewards is "safety" but another they will work for is food. ie if you go out every day and feed them off the back of the ute, they will follow the ute to the next paddock for their feed - and you don't need a good sheep dog.
Applying an aversive and then removing it when the animal does what you want - is negative reinforcement - but I prefer to call it "escape training" ie the animal escapes the the aversive by doing what you want. This is how prong collars, choke collars and zap collars should be used. It is also why you need really really good timing. Eg with the choke collars - and this is often how I see them used - if you never release the pressure the dog never learns what you want. A lot learn to ignore the pressure and you damage their trachea long term.