POST and
BIST meaning depends heavily on the application domain. Generally they are meant to identify a design behavior that deviates from the fault-free operation, because of external (perturbations, limit/unsafe operating conditions, ...) or internal (manufacturing problems, faults, ageing, ...) causes.
Please note that the "fault-free" term doesn't indicate that the design/device performs correctly, as the designer intended. Bugs or implementation errors may cause the circuit to behave differently from the stated goals/specifications/objectives. Design Validation is intended to catch any problems during the design phase. Various techniques are available: simulation, formal verification, etc. The approach usually consists in comparing the current design/implementation to a reference (i.e. judged to be fit) model, identifying and correcting the errors and so on.