While these may make sense to some, none of them are their intended purpose. To put it in perspective, think about what that hand is doing before it gets to your belt...it's out in front of you doing something: punching, blocking, grabbing, etc., and then you're pulling that hand to your belt. This motion is most evident when performing the basic blocking motions (actually, strikes, remember?) that we learned from our first days of training: the high block, low block, middle block, knife hand block, etc. In other words, one hand grabs the opponent and pulls him toward you while you execute the block/strike to amplify it's effect.
The pulling hand is also used for many other reasons, such as pulling to gain leverage on your opponent, unbalance him, to set him up for a follow-up technique, or to throw him altogether. Many katas show these techniques, probably none more so than the Naihanchi/Tekki/Chul Ki katas. For my PaSaRyu brethren, a perfect example is the second move in our yahk soak deta (promised sparring) 1st set number 2, when the opponent's punching hand is pulled toward you.
Now keep that hand at you belt!