I think what most people mean when they say docker-in-docker is in layman terms. You simply just a way to do "docker run" from within another container. Whether that's done by actually running docking daemon inside or mounting something from the host or whatever else i don't really care. I just want it to work.
Sure, there are some details you might want to control, like image caches and such being shared with host, i just find there is a lack of documentation and best practices of how to do nested docker, if that is even intended to work or if mounting docker.sock is an unsupported hack. Most information found about this is scattered on shady blogs.
For the examples of CI brought up the use cases are fairly obvious, you have a jenkins installation with x plugins installed - running as a container. Within this jenkins you are building multiple different projects which all require their own respective image to build. As a project developer here i don't even want to know if this jenkins is a bare metal, a vm or a container. Here docker is more used to bundle all the dependencies, not for strict security with perfect containerization.