The instructions are not clear enough - a bad translation - and they are incomplete. Only after wasting quite a lot of time and effort and materials did I discover the cause of the problem: badges were not 'coming together' properly. Was it my technique? Or a fault in the equipment? Or in the materials? Eventually, on YouTube I found a badgemaker who had experienced the same problem, with the mylar disc/paper disk not being 'crimped' into place and falling apart:
Vevor supply a plastic ring that sits in the second press beneath the badge back, and suggest you use it with the metal badge back. However, this is/was not enough - you need to lay a metal badge back (with the pin removed) on top of the plastic ring. Then when you lay your badge back on top of that, it's the right height. I don't understand why this information wasn't supplied with the product - so, a lot of wasted time, energy, and materials - but now it works fine.