You are right the core of most optical fibers is either 8-9um for singlemode, or 62.5 or 50um for multimode. The cladding, which is also made of glass, surrounds the core and this is almost always 125um. Often there is more than one layer that makes up the cladding glass to help reduce the bend radius before you start to attenuate your signal. You need both the core and the cladding of different refractive indexes to create total internal reflection, which is how fiber optics work over long distances with low loss.

The glass (core + clad) is the only part of the fiber that is really recyclable. Everything else is plastic that is difficult to chemically remove.

There's a lot of really bad literature out there on fiberoptics, so I don't really blame anyone for not knowing this stuff. Here's a pretty good article that sums up how fiberoptics work I pulled off google: https://www.ofsoptics.com/faq-guide-to-fiber-optic-cable/how-do-communications-fiber-optic-cables-work/