Those are different concepts: the point of approximation and the point where you evaluate the policy functions. It is almost always the deterministic steady state around which you do the perturbation, while the point where you compute the policy functions at higher order is either the stochastic steady state or the ergodic mean.
What exactly is it you are trying to compute, i.e. what is the idea of the exercise?
Regarding the old post I forgot to answer:
- Depending on what you are doing, second order is sufficient. If your simulations do not explode, do not use pruning. The pruned state space implies that you are actually solving a different model. We only do this if necessary.
- Yes, it is the uncertainty correction, i.e the precautionary behavior that drives you away from the deterministic steady state, even if no shocks happen.