As far as I know the most plausible explanation previously offered was that they all started with concentric circles and then a 'polishing process' (presumably some sort of dipping) took place and depending on if and how long this was done for the circles would disappear.
Now, as I didnt actually believe that at all, I broke open a number of CIC's and seperated the discs to analyse it further. This included talking to a friend at a metalwork factory.

I concluded :-)
The discs with concentric circles are mild steel and were cut from a circular rod whilst the rod was spinning (presumably in a lathe or similar)

The discs without circles are an alloy punched out of a sheet.

I do not disagree that there was a period in time when both were being produced. I put this down to one type being made in the US and the other in Mexico. I reckon Mexico would have started later than the US and produced the alloy ones as by then they would have figured out it was more costs efficient for a new plant to make them that way.

Things noted along the way -

If the disc has concentric circles on one side then it has them on both sides. If the disc has no circles one side it has no circles both sides.

Discs with circles (steel) are heavier.

Discs with circles have a clean polished edge (because the bar they were cut from was polished by nature)

Discs without circles have a rough slightly bevelled edge where they were shorn through by force of a die on a sheet of metal.