*A points are awarded based on the actual operating carrier, irrespective of which airline code you've booked under. So it makes no difference in your case as you wouldn't get anything with Airpoints on the EI-operated segments.
NZ60 is SYD-RAR-AKL and NZ69 is unused, so these would be EI60 and EI69 respectively. Likely a bug on the 3rd party's end.