Are there different "down" commands for the GSP?

From what I read on the German DK forum, "down" (English command) means the dog needs to drop,with the head down as well, not head up looking around. Apparently, the German "platz", in Germany is reserved for formal obedience and not used with hunting dogs? And it sounds like "down" is a very serious command as it is used to drop the dog to the ground when needed, in the field. Again, dog is expected to drop body and head to the ground. It is different from the command given for "ablegen" e.g. telling the dog to lie in one place until released ("platz".

I can see that can come in handy. What does "down" mean to your dogs? Do you teach them different ones? And how?

For me down is down.  I teach a moving down, a down on recall, a down stay, etc., but they are all down.  The only time I want their head down is when they are in a down stay and only because they are less likely to get up when their head is down.  I have never taught a special command for that.  They are smart enough to know that a down stay they might as well get comfortable because they will be there for a while.  I prefer my dogs to fold when they go down, meaning head and front go down first and then back, but some of my dogs just don't like to do it that way and there is really no harm in the sit then down unless you do agility.  Of course the down on recall is easier and looks better if they fold down.

I use off in the same way as the others.  My dogs have all just picked up the relaxed obedience down stay.  Meaning when I down them at my side for the long down I start out telling them to get comfy and gently flip them over on their side so they aren't sitting like a sphinx.  They eventually do that automatically and I don't have to continue the get comfy command.  I just say down and they know by my position which down we are doing. They also automatically put their head down on their paws, so I have never really had to teach that in the obedience down stay.  On the drop on recall or moving recall my two youngsters automatically did the fold, so I didn't have to teach it.  Belle was the one that would not do the fold and I think it was because she hurt her front shoulder and had to have surgery when she was about 9 months old.  Off means get off of whatever.  Four on the floor is my no jumping command.  I generally only have to use that when they are younger.  Later they learn not to jump unless invited.  My jump up and put your paws on my shoulders is Hugs and Kisses. 

When you treat her for the down do not release her right away.  Wait a few seconds and praise her for a good down stay, then release. 

I trained Ringo by luring as well, but quickly faded the treat lure.  Then we worked on lengthening the amount of time he was down.  At first I stayed very close so I could reach out and correct if he tried to get up (I also tried to anticipate him based on his body language, like dropping his head).  We started with just a few seconds and worked up from there.  I have now progressed to moving about 12 feet from him and we are up to about 2 minutes at that distance.  Often if they are popping up at this age it is an insecurity issue.  Try moving back closer to the dog. and rebuild.  Also, he doesn't get  a treat until the exercise is over and I have said my release word.  Then he may get up and he gets a treat.  This proofs for when you are in the obedience ring and you move back to the dog and the judge says exercise over.  You want to be able to reach and connect the leash and then wait to find out if you qualified before the dog jumps up.  So, my dogs have to wait for the release.  I am now at the point with Halo and Belle that if I say stay and start walking away and get to the end of the leash and tug, they do not budge.  They are waiting for the release word or the word heel.