Hello, idkid:
If you notice, to some extent the grip in the two videos is similar: the wrist does not move and the fingers pivot the razor on the return. The grip cannot be the same in the two versions, because one is used for a hanging strop and the other for a paddle. The principle, however, is the same.
Please note that like anything else in the world of wet shaving with the straight razor, stropping also is influenced by personal preference. There are no set rules. What matters is the optimum method the shaver chooses for him or herself.
For me, although I do not own a paddle strop, the style AFDavis employs is ideal. He has an excellent style for the paddle strop and gets the job done well. If I were to use a paddle strop, I would emulate his style.
I use a hanging strop and my style is similar to hi_bud_gl's. That style is optimal for me as well comfortable. It accomplishes everything my humble razor expects of my strop. There are no wasted movements, and the job is done well.
Other gentlemen might prefer their own stropping styles, and recommend them. That's fine. Choose the method that works best for you. When I recommend a stropping style to newbies, the two in the videos noted are my selections. Other veteran shavers might suggest other style. That's fine, too. It's personal.
The two videos noted are excellent in every respect and I recommend them. I also thank the gentlemen who made them. Good stuff.
Regards,
Obie