We study the influence of delayed optical feedback from a short external cavity (EC) on the emission dynamics of vertical-cavity surface-emitting semiconductor lasers (VCSELs). We find the emergence of pulse packages (PP) exhibiting characteristics which originate from the interplay of time-delay induced dynamics and polarization mode (PM) competition. We provide detailed analysis of the polarization dynamics applying complementary cross-correlation and spectral analysis techniques. The analysis reveals an interplay of the dynamics linked to the hierarchy of time scales present in the laser system: the fast time scale of the EC delay, the slower time scale of the PP, and the slow time scale of the polarization mode competition. For an analysis of the dynamics we provide a toolbox of methods adapted to the different relevant time scales and temporal variations of the dynamics. This complementary view unveils distinct changes in the relative oscillation phase of the two PM for increasing the injection current on both the fast and the slow time scales. Our results emphasize the significance of the polarization mode competition for PP dynamics in VCSELs, contrasting the observed behavior to what was reported for edge emitters.