Homemade Plectrum Punch

[Toolmaker51]

brianhw; that qualifies 100% as die set. I'm impressed.
While swiveling is not common, the stiff hinge controls alignment correctly for punch to enter die 'centered'. Normal dies slide on pins and bushings. Your die is a blanking tool. Item sought is plectrum, remainder is fall off [scrap]. Piercing dies are opposite; item sought is surrounding material, remainder is scrap, often called the slug.
In either case, punches can be angled to produce shearing action. That angle can generate one or more shear edges, when center [peak] cuts first those are called 'rooftop' punches. The spacer provided in yours makes it more parallel. You mention the hand filed perimeters fit 'so-so', but the die side does far more to control resulting shape and size than the punch. Gap between punch and die is called 'clearance', and is adjusted primarily according to material, thickness, and total length of part profile(s). Those lead to calculations of tonnage needed to operate the die for parts Most of us describe the space in thousandths 'per side'. Something like a washer might have .012 per side; both around ID and OD. A larger plate of same material and thickness would be spaced equally on each edge, and need more tonnage.
Following site link has descriptions of die work elements and basic tooling, I've posted it before, as one of my favorite references. https://www.roperwhitney.com/our-products/
This one has Ms. Toolmaker51 re-dedicating work to decorative work. The physics are identical. http://www.jayneredmanjewelry.com/jayne-redman-tools-2/