I think we can get a plurality to agree that a Possibles Bag is a distinctly different bag than one would hang next to their gun. I always saw a possibles bag as one that you would carry to town, to the neighbor’s, or on a quick overnight stay somewhere. It would be slung with the hunting/shooting pouch if walking and probably attached to the saddle or thrown on the floor of the wagon so what you might need to have to hand, outside of your small number of shooting accoutrements, was accessible when away from the house. We have to remember that everything was carried everywhere. Think about the stuff you throw in your truck when you go here or there. What you carry when you go for a walk, go fishing, or go on a hike. All these things were carried in bags, and I'm certain even the poorest had two to three different bags on a peg, preloaded with the basics for this trip or job, to be added to before walking out the door for that particular day.... Haversack, day bag, fishing bag. If on a farm, a bucket of tools.....
These thoughts are based on living and working full time on the farm, where every day has me loading the toolbox on the tractor or putting this thing or that in the front rack of the 4-wheeler for what is several hours of work up to a half mile away from the house. I keep the basics on the tractor or the 4-wheeler but that is just the core of what I will need for my day which could be any of a hundred different places or jobs. If I was walking a mule from the barn to my worksite every day there would be a set of bags hanging in the barn to throw over ol' Stubborn's back with my sammitches, water, and whatever specifics I needed for what I was doing. This was day to day life back when.... One bag for this, one for that, and a Possibles Bag for what you always need, regardless of what you're day is going to look like, work, calling, or just going to take a nap under your favorite Hickory tree while momma thinks you're plowing the back 40.
So to answer the OP, I think the proper name for a 'Possibles Bag' is Possibles Bag, among others. It had the same function as a ladies purse.