We do not consider the Cook Islands to be a country but it is included in the "Countries 197 to 205" quiz. It is common to include these countries but you should always indicate this in the quiz description. E.g. "Countries displayed in red are from the "Countries 197 to 205" quiz".
I do, but it's not a requirement or even a convention to do so (if you do, that nearly gives the game away because there are only 9 countries to choose from in Countries 197 to 205). Why are the Cook Islands a separate country? Because when you fill out the NZ census asking for your country of birth, the Cook Islands are listed as an option! Everyone in NZ, including the NZ government's own Department of Statistics, considers Cook Islands to be a separate country. That census question answer is the source data for this quiz.
Interesting debate and thanks Jerry for your explanation as well. I've put a small caveat in the instructions. I think this is enough to keep most people happy.