I think some of the differences of opinion have to do with two markedly different activities, that are both called "lunging" , but are really two different things, IMO.
The traditional manner of lunging has the hrose quite some distance from the handler, responsive to verbal cues and is more likely to work in one direction for more time before turning. And turning around is done slowly and by the direct contact of the handle with the bridle, as they may detach and reattach the lunge line to support the new direction. So, it is advisable for the hrose to stop ON the circle and await the handler. The objective is more to get the hrose moving forward, listening to the verbal commands, to maybe loosen up and let off some steam, and with the use of side reins, to perhaps encourage the hrose to learn to soften at the poll and perpare him to carry himself better under a rider.. the lunger strives for a steady rythmic gait, and is not interested per se if the hrose is "asking" to stop or not. They are looking for a willing forward and enough steady forward gait for it to become stable.
The other type of "lunging" is more associated with Natural Horsemanship. This will have the hrose working much closer to the handler, working more on the horse's attentiveness to the handler, as evifdenced by its' willingness to change directiions when asked and be ready to stop and await further direction. The "nuetral" position, wherein the hrose is awaiting further instruction, is for it to be facing the handler. From there, the handler can either back horse up, draw it in, send it back out onto the circle right or left. The sensitive arrangement of these direections will tune up the horse's sensitivity to the handler.
Also, as the horse does walk around the handler, on a much shorter line, the handler may ask the horse to disengage it's hind fully, to loosen the pelvis and test the "brakes", or, may ask that the horse walk around with a good bend in its' body and clearly stepping under with the hind to maintain this bend.
So, less trotting or cantering for circles in repitition is done. Different kind of feel, different kind of outcome, maybe.
I dont want a horse to turn and come in to me when lunging. He can stop and turn toward me, or better yet, turn his head inward, but not come toward me until I invite for THAT.
A lot of what i do in lunging, and I am NO expert, is more toward the natural horsemanship way. Though, i will ask the horse to trot around and just look to see it move out and loosen up. but I will also work and getting the horse to feel of the lungleine, and if I aske him to look in, then I want to see him follow the line (represents the rein) inward, and maybe follow it all the way around to a stop. I certainly do not want to spend my time with a hrose trotting around endlessly, leaning on the line , stiff and counter bent, which is what I see a lot of in lunging.