NJ Sportsman,
Ignore those suggesting you should know this already. We all learn somewhere and don't need to pass their litmus tests for whether or not a question is legitimate to post or not. Some manuals do explain the riming systems, but not all. I was also disappointed by the Wikipedia article on the subject for having poor illustrations. I may undertake to correct that.
Both Boxer and Berdan primers have a metal cup with a dried pellet of priming mix in the bottom. In both instances, the cup is inserted into the primer pocket of the case. In both instances, the mixture is not rigid enough to be fired by merely striking it with a firing pin on the outside of the cup. There needs to be something to brace the mix against for the firing pin blow, such that the pin can squash the mix between itself and this support. The support is called an anvil. Where the two types of priming differ is in the way the anvil is configured.
Boxer primers have anvils built into them. The anvil is a little brass object with either three or two feet with wide pads on the ends. When the primer is seated, the anvil's feet first touch the bottom of the primer pocket and then, per specification, the cup is driven in about 0.003" further to ensure the priming mix is solidly supported over an area. That is called reconsolidation of the primer or setting its bridge. As the feet of the primer cover the outer perimeter of the floor of the primer pocket, the hot gas vent (the flash hole) is made in the middle of the pocket of a Boxer primer and the hot priming gases evolved escape between the legs of the primer and into that hole to get to the powder space in the case to ignite the powder.
Berdan primers have no anvils of their own. They are just the cup and priming mix, which is why they are less expensive to make. Instead, the anvil is formed into the brass case as a bump in the middle of the floor of the primer pocket, something that takes no more effort than forming a flat bottom primer pocket, and the Berdan primer is seated against that bump for reconsolidation. Because the bump is in the way, the gases cannot escape through ha hole in the middle of the primer pocket floor, so either two (most common) or three smaller holes are made around the perimeter of the bump at the bottom of the pocket, and the primer gases escape into the powder space of the case through these.
Obviously, when you have a decapping pin that is in the middle of your die, it will run into the inside bottom of the anvil in the Berdan system primer pocket instead of into one of the vent holes. The only common ways of removing Berdan primers are a special dual pin punch, a can opener-like device that pierces and hooks the spent primer cup out from the outside, and the use of hydraulic pressure, done by setting the cases into a holder that is open under the primer, like a shell holder, filling them with water or oil and pressing or hammering a snug-fitting dowel or metal rod down into the case mouth and hammering on it or pressing it up against a solid stop rather quickly to force the fluid through the vents and behind the primer cup to blow it out. In short, it's a bit of a nuisance to do.
When you have cleaned out the oil or dried out the water from a decapped Berdan system case, you can seat a fresh Berdan primer into it if you can buy one the right size. That's another problem. There are more different diameter Berdan primers than there are Boxer primers for common small arms cartridges, so you have to get the right one.
It is all very slow work, and unless you have a rare chambering for which no boxer primer cases are available, it hardly seems worth the extra bother.
The cases you have are most likely Eastern European or Russian. I say that because, being marked .223 Rem instead of 5.56 means it was not military ammunition, but it is not uncommon for such cases with Berdan priming to be turned out on contract by a factory in one of those countries, as they do a lot of such contracting and they can help keep costs down using Berdan priming.
Interestingly, Berdan priming is a U.S. invention and Boxer priming is a British invention. How we decided to prefer their system and they, ours, I don't know the history of.
