I've just been through an extremely painful exercise of this with Air New Zealand.

You will get a fully flexible airfare (and airpoints in the full economy category) when you tick the checkbox and book The Works or Works Deluxe (Booking class M,B,Y).

Not ticking the checkbox will give you the cheapest currently available booking class for the selected product, e.g. for The Works and Works Deluxe it might be LTWV (no airpoints) or QH (economy discount airpoints) or even MBY (full airpoints).

The problem with the engine is that you can't choose to get a Q/H ticket if there are cheaper seats available. In that case you have to call and book via the phone. Stupid. I wasn't charged a fee for doing that btw and the CSR on the Gold Elite hotline I finally ended up with said they wouldn't charge the fee for that. YMMV depending on the person you speak with.

The story seems to be different when it comes to Seat + Bag (and probably also Seat). S+B doesn't seem to have the equivalent of a fully flexible economy ticket. The booking classes available there are P,K,X,G (no airpoints) and S (discount economy airpoints).

If you ask me, AirNZ has messed this up for a lot of their HVC and frequent flyers. The new fare structure might be good for the tourist taking an occasional trip to between Aussie and NZ and can now choose to save 20$ to not have a bag but for a lot of people spending significant amounts of money it's just become a pain in the ****.