My thoughts on full-stack is like your thought. It's a developer position. Once you start throwing in tasks that include the word "administration" in the description then that's likely a different job for a different person.
I joke that I'm a "full full stack" developer. Writing JS (bare, or in a framework like React or Angular or whatever), backend (Python, Elixir, Go, Ruby, etc...), install the OS or design cloud infrastructure, attach GDB or ptrace to processes if they're misbehaving, use tcpdump to watch packet flows, program kernel drivers, put together a toolchain and JTAG for microcontrollers, and design schematics & board layouts (analog or digital, not great at RF... yet).