"....is it still desirable to keep a planted pivot foot during a horsemanship turn, or is it more acceptable considering the increased speed to have that foot raise and lower in the same place or even have both rear feet changing position but keeping the hip more or less in the same place? ..."
It is not only desirable to keep the pivot foot planted, but it is the correct way to perform a pivot. When executing a pivot, also known as turn on haunches, it is never acceptable for the horse to raise the 'pivot' leg. That is the hind leg that would be on the inside of the turn. The front legs must cross in front. If you are going to execute the pivot with speed to maintain your pace throughout the pattern and your horse lifts the pivot leg, you will be counted off. That is another reason to remember that just because you can do one maneuver swiftly with precision, does not mean keeping up that pace throughout isn't going to cause you problems with other maneuvers you may still be finishing in training at home.
Most horses that are being trained for western classes that are not classified as speed classes (i.e. western pleasure, horsemanship, etc) usually learn the pivot in hand as it is also a maneuver acceptable in showmanship patterns.
Remember if the pivot leg lifts, it's a fault. To what degree it is penalized on the final score is up to the judge. If the front legs are crossing in the front and not behind one another and the pivot leg is only lifted once, possibly twice if the footing is grass or firmer ground (it's easier for a horse to pivot on loose footing, sand, sand clay, crushed stone, etc) I would probably only count down a point but that really depends on many other factors.
When practicing pivots at home, it is wise to use hoof and leg protection whether practicing in hand or under saddle.
More from this readers' inquiry yet come...!