The blue one with the yellow lid is the old recycling bin and they gave us a new and bigger one (green). You can now either use the old blue on for rubbish with the tag, or buy the pre-paid rubbish bags. Bin + tag is cheaper I think.
Looks like they are in Auckland and you've linked to the Wellington page. But likely it works in a similar way.
It's also possible you can get a rubbish wheelie bin from a local company. I'm not in Auckland but normally the cost of collection is paid for in when you buy the rubbish bags, so you aren't paying twice if you use a wheelie bin from a private company. They also often let you choose a smaller bin, have fortnightly collection, and other options like that.
We have council wheelie bins here and pay for tags for each collection.
I'm guessing they don't have a problem with people using "recycling" wheelie bins as rubbish bins when clearly marked with a rubbish tag, as that must've been what the previous owner was doing.