I think my two star chips are the same as the less embossed chip you had in your first post. I added a monarch-mold (raised circle mold) chip just as a reference.

If you look closely at the outer rim of the white star chip, you can see that it is slightly discolored and more worn. This means it was a "live" chip, like the monarch-mold. The Monarch-mold was advertised as the first "live" chip in which contact between chips was only made on the outer rim. Early 20th Century gambling catalogs advertised live chips as being superior because the minimal contact around the edge of the chip made stacks of chips less likely to fall over. They even claimed that the space between chips created a vacuum that held the chips together, although I somewhat doubt the physics of that.