I know of at least 7 terms used to call an opalescent variety of chrysoberyl, which are: Chrysoberyl Cat's-eye, Cymophane, Cymophanit, Cymophanita, Cymophanite, Cats eye and Cat's eye.
Which one should I used, why and where?
The common trade name for chatoyant chrysoberyl is cat's eye. or chrysoberyl cat's eye. Chrysoberyl can be chatoyant but not opalescent. Other cat's eyes like tourmaline or kornerupine should be reffered to as tourmaline cat's eye or kornerupine cat's eye respectively while chatoyant chrysoberyls can be designated as cat's eyes without the species name. Cymophane is another name for chatoyant chrysoberyl but is rarely used in the trade.