Well in that first link, the instruction PDF does state to "wipe the lens with smooth circular motions of the cleaning tip". From this I would say that the tip should be moving in a circular motion, the movement of the body is not important. Personally, I spin it the same as I would spin a coffee stir stick. Spinning the entire body on it's axis is quite difficult for my wrist, so this is the only logical way I've found to do it.

I think that a straight line would be just as effective, but since most lenses are convex, the circular motion just makes more sense.

I will point you to a thread about the application of car wax at Meguiars Online. In essence they point out that when applying wax:

If you’re applying non-abrasive products, i.e. products that do not scratch or scour the finish, and your applying them with a soft, clean applicator and using good technique, then you should not be instilling any scratches or swirls no matter what direction you’re moving your hand.

I believe that the same applies to the lens pen. You could try to use straight lines, but the circular motion is more natural for most people.