As it is generally used today, a "tinderbox" is the container you carry your flint/striker/char in. And most things being sold as "tinderboxes" are actually tobacco tins - like those oval brass ones with a magnifying lens in the lid. Those are "tobacco" containers.
A true tinderbox held bits of charred punky wood in it. Sometimes your flint/steel were also kept in there. You struck sparks down into that "tinderbox" until that charred wood caught a spark. You then fished that ember out and started your fire. And you also then put the lid back on your "tinderbox" to smother out any embers still burning in it.
A lot of the early "tinderboxes" were true wood boxes - carved out of a thick board with a fairly tight fitting lid to set down in them. That tight fitting lid put out any still burning embers inside it, and saved that charred wood bits for the next time you needed to start a fire. When you needed to refill your "tinderbox", you fished some coals out of your current fire, put them in that "tinderbox", and closed it up. Those coals would then smother and go out - and be ready for you the next time. The wood tinderboxes usually ended up pretty "charred" or blackened on the inside.
The metal "tinderboxes" worked the same way - just in a tin container instead of wood. They often had an extra plate to set down inside to snuff out burning embers. And often had a socket on top for a candle. You stored your flint/steel inside but on top of that snuffing plate. So you had everything in one container to start a fire and get that candle lit.
Those pistol shaped items were just a mechanical flint striler. It worked just like the lock on your flintlock gun. But the "pan" was often very deep - to hold your charred bits of wood to catch the sparks. And many times, the "barrel" was hinged and a candle was stored inside it. Sometimes that candle was already in a socket that swiveled up, other times it was just stored there to be taken out and set down in a socket attached to the "pistol" somewhere on the side. They usually also had a pair of legs under the "barrel" to hold it upright. CVA or Dixie used to sell kits to make into a small one of these. I picked one up at a garage sale years ago, but I need to get or make a trigger for it. The rest is all there. One of these --- projects ---- to tinker on sometime.
So those oval brass containers are actually Tobacco containers, but many people use them to hold their fire starting kit. They work very well for that.
Just my humble thoughts to share, and best used in conjunction with your own research.
Mikey - that grumpy ol' German blacksmith out in the Hinterlands