I checked the NYS form, and there is an “Independence” party. The no-party option is “I DO NOT WISH TO ENROLL IN A PARTY”. However, I am sure that Mr. Wildwoman and everyone here who has referred to being registered independent means that aren’t registered in a party – it’s all pretty clearly marked. I assume the vernacular was formed as ‘registered independent’ before this party was around, and it just didn’t change. And the voter ID would say "Independence Party" if it meant that, so I suppose the official wording was just never changed.