Thats not really the definition of "run" Pixie, a dog may have "run" and be you can be able to keep up with it on foot , you are describing the "run" required of an all age dog or horseback FT dog ..
A foot hunters dog can have "run" and move with a similar fashion and speed in a closer area . Yor dog should have "run" to do well in Foot Field trials as well ..
Its all about the style of the dogs movement . A field dog can have
"run" but it searches every bush in the field this dog still runs with the style but would be considered a "meat hunter" and not a good FT dog ..
Thats what i said its hard to explain , someone gave you a simple explaination you could understand easily without seeing it ...also Like i said I aint all that familiar with Field trials but i am somewhat familiar with dogs with "run" and dogs without..

i would say this is the definition" being wide open all the time" the rest of the definition given is basically refering to the run required for a specific venue I would guess to make it easier for you to understand without actually seeing it ..

Your dogs or a show dog may have run , someone would need to see your dogs run in the field to know for sure, but "run" is not a requirement of a show nor is a requirement of a hunter nor is a requirement of probably most DK breeders. therefore most could probably care less about breeding for it .. Some do as it is a very nice style to see in the field, most probably dont or probably could care less ..
Most FT people probably do breed for it as it is a prefered style by most FT judges from what i understand also a dog with "run"  is a dog with great quickness as well..
There are many other styles that are just as beautiful to watch in the field in my personal opinion..

 Ok, now it is my turn to disagree with a post.  Briar you said

""run" is not a requirement of a show"

I beg to differ.  Good movement and conformation are a requirement of a show dog.  The difference is we can only display that movement at  a trot in most show venues because of the ring limitations.  A good show dog though should have that excellent movement at any speed and should be able to keep it up for hours.  Just because some judges are not very good, rings do not accomodate the gallop (my term for run based on dog physiology) and some show dogs do not have it does not mean it is not required.

Here is an excerpt from the standard. And yes I understand some is subject to interpretation, but I would maintain that a dog who has poor movement at whatever speed has conformation issues, and would not do any better at a gallop or run.  And likewise a dog that fits the description below and has excellent movement and conformation at a trot will be able to move just as well at higher speeds and keep them up for extended periods.  And yes we then get to the debate of whether said dog could hunt, but that is not my point here.  I am only talking about movement and gallop or run.

"General Appearance
The German Shorthaired Pointer is a versatile hunter, an all-purpose gun dog capable of high performance in field and water. The judgement of Shorthairs in the show ring reflects this basic characteristic. The overall picture which is created in the observer's eye is that of an aristocratic, well balanced, symmetrical animal with conformation indicating power, endurance and agility and a look of intelligence and animation. The dog is neither unduly small nor conspicuously large. It gives the impression of medium size, but is like the proper hunter, "with a short back, but standing over plenty of ground." Symmetry and field quality are most essential. A dog in hard and lean field condition is not to be penalized; however, overly fat or poorly muscled dogs are to be penalized. A dog well balanced in all points is preferable to one with outstanding good qualities and defects. Grace of outline, clean-cut head, sloping shoulders, deep chest, powerful back, strong quarters, good bone composition, adequate muscle, well carried tail and taut coat produce a look of nobility and indicate a heritage of purposefully conducted breeding. Further evidence of this heritage is movement which is balanced, alertly coordinated and without wasted motion.

Gait
A smooth lithe gait is essential. It is to be noted that as gait increases from the walk to a faster speed, the legs converge beneath the body. The tendency to single track is desirable. The forelegs reach well ahead as if to pull in the ground without giving the appearance of a hackney gait. The hindquarters drive the back legs smoothly and with great power."

At the end of the day, I think the path to betterment of the breeds lies in breeding for an all around good dog and not focusing on one particular aspect of the dog.  If all we worry about is gait, surely other aspects of the package will deteriorate from the original intent of the breed.  Likewise if we only breed for a pretty dog other areas suffer.  It is in all our best interest to look at the whole dog and breed accordingly.  I know most of you do not think very highly of the JH, but I would be a strong proponent of requiring a JH on a dog before it can be shown in the breed ring.  That at least insures the basic hunting instinct remains in the dogs that are bred.  I would also love to see some type of conformation evaluation/test on some of the field dogs prior to seeing them bred as well.  And while we are at it a temperament test too.  I have seen far to many dogs in various venues that are being bred that I frankly wouldn't own because they aren't the whole package, and I don't care how many titles the dog may have before and after his name.