It depends on which company and their application. At the regional/135 freight level, most allow both forms of PIC time to be reported. However, at the major level, they might want to only see PIC as defined in part 1, whomever is the actual PIC, not just by loggable definitions. I would still log post Private training as PIC and dual received because you can always make an adjustment for it on the application and only the airlines seem to want that specific time.

Logging whatever you can is good in some instances such as corporate jobs whose insurance companies require specific requirements.