Are business case & project initiation document roughly the same thing? Do they fall under the initiation stage or the planning stage of the project life cycle?
Thing of the business case as being primarily cost v benefits.
The PID will include the business case but will be more comprehensive. For example, it will include timings, responsibilities and personnel. Think of the PID as the ‘handbook’ of the project.
I would out BC under initiation because if that doesn’t stand up then you won’t start planning. PID can be the the output of the planning stage.