This is known as the PID-1 issue, and ever since Docker first came out it's been a major source of frustration for many users who want to use Docker containers to run their existing workloads.
It's not possible to run a Docker container with systemd, upstart or sysvinit working without major workarounds. So while it is possible to use a custom and hacked environment to get a docker container that simulates a regular linux system, out of the box it's simply not possible.
I think it's entirely accurate to call Docker a system for managing application containers. LXC can be run with various options, but by reading the lxc docs (one example here: https://wiki.debian.org/LXC) you can see that it supports various normal init systems and not just the process managers that can run under Docker.
It's not possible to run a Docker container with systemd, upstart or sysvinit working without major workarounds. So while it is possible to use a custom and hacked environment to get a docker container that simulates a regular linux system, out of the box it's simply not possible.
I think it's entirely accurate to call Docker a system for managing application containers. LXC can be run with various options, but by reading the lxc docs (one example here: https://wiki.debian.org/LXC) you can see that it supports various normal init systems and not just the process managers that can run under Docker.