As I'm currently pursuing an academic job search (in parallel to the industry one), which includes potential postdoctoral positions, I would appreciate some clarifications on the following aspect.
Freedom. I assume that there are two major options for postdoctoral positions in regard to academic (research) freedom: 1) postdoctoral researcher joins a group, lab or project and contributes to one or more of their research projects under PI's guidance; 2) postdoctoral researcher joins a group, lab or department and contributes to one of those entities' research portfolio by conceptualizing and performing her/his own research (still under PI's guidance), based on previously negotiated agenda, which fits and supports larger groups' research themes.
Question: are my my assumptions correct; does a postdoctoral title reflect the type of freedom (or lack of) of the position?
Note: My field is Information Systems, a discipline, which can be considered as a sub-domain of Management Science, which, in turn, is a social sciences discipline.