Should I add a vapor barrier in this situation?
Foam board, especially the one in your pics, act as a vapor barrier.  The only issue with it is when you cut it and install between joists or studs.  Then it needs to be taped, caulked, or foamed.  You said you caulked them so you have completed your air seal. Adding another vapor barrier will only cause moisture problems and wood rot.  I have been insulating homes for over 30 years and have seen the entire sides of homes that have had to be tore out and rebuilt because of the second vapor barrier.