That looks interesting, but it does not appear that it would hold enough to suit me. I run to and/or from work and I often pack breakfast, snack items, or some item that I may have forgotten to bring with me when my wife dropped me off with my week's worth of supplies on Monday.
My pack is an Eddie Bauer pack that I bought quite some time ago. It has three pockets. A main pocket that is fairly large. A smaller outside pocket that I always leave empty now because the zipper is broken. It also has a flat pocket with an outsize zipper. That flat compartment sits right against my back and if I don't have enough stuff to fill the large compartment, I will put it in that flat compartment and it will be held snug against my body with minimal or not bouncing. When I am only using this pocket there is very little bouncing. The empty part of the pack probably bounces a little, but since it is carrying no weight, it is not noticeable.
Nevertheless, I do wear the fanny pack snugly to prevent bouncing. The heavier I pack it, the more snugly I have to wear it. When I am only using the flat pocket, it does not have to be quite so snug because there is little weight and all of the weight is held close to my body.
From appearances, the running buddy might work. If it were just ID and keys, I would think that it would work fine, but I think that an iPhone will make it heavier and more prone to bouncing (and perhaps dragging your shorts down).
You might need very good elastic on your shorts or to tie the drawstring fairly tightly which would defeat the purpose.
I have run with three different fanny packs and with a lightweight, framed backpack. From my experience, I find that I always need to have my pack secured snugly to my body. The lighter the weight and the closer it rests to my body the less snugly I need to secure it to prevent bouncing. Also, if you're using a belt or fanny pack, a wider belt makes the snug fit more comfortable.
If you could find some compression shorts with a pocket in the middle of the back that is large enough to accommodate your iPhone, that would probably be the least bouncy option available. I have some compression shorts with such a pocket that I wear for long runs when I am on call for work. It carries the cell phone with absolutely no bounce, but my particular pair of compression shorts would probably not accommodate an iPhone. The smaller, non-smart phone that I carry for work is a tight fit.