In a client that is set to a STOPPED status, some tasks such as jobs and scripts can still run. There seems to be conflicting information in the documentation around what the expected behavior is.
The described behavior is as expected - when a client is stopped, any task that has a parent object (superordinate task) will go to a "STOP - Client-wide stop of automatic processing" status (including but not limited to workflows, events, schedules).
However, if you run a single object like a script or job that does not have a superordinate task, these will run.