First of all, it is very common when firing primed but not loaded brass in a revolver for the primer to back out of the pocket and "tie up" the cylinder. The gases from the primer are restricted going through the flash hole and cause the primer to back out against the recoil shield. The same thing happens when you fired loaded rounds, but the pressure in the case causes it to back up and reseat the primer. So you just experienced what is common.

The cases with the ring, don't appear to have been crimped, but!!! in many cartridges a circular crimp is used to retain the primer. But I have never encountered that on 357 Mag brass. 38 Special military brass, yes.

As a common practice, I run lots of brass through my RCBS primer pocket swager. I do this with all crimped brass, and brass with tight primer pockets like S&B. So if it were mine I would just swage the pockets and be done with it. It puts a nice radius where the "ring" is, and insures the pocket diameter is uniform and not undersize.

Don't concern yourself with the primers backing out, because they do every time you fire them.