What's in a name?
I believe gyroplane is the word used in the ANO and by the CAA, which appears to be a generic term to describe all flying machines of that category. Gyrocopter and Autogyro are particular company's tradenames for their own versions or models of gyroplane, which have come into common usage. Gyrocopter is used by Benson and Autogyro (or Autogiro) by Cierva.
Similar to other commercial examples such as Hoover/vacuum cleaner (for those of us old enough to remember the pre-Dyson years), JCB/excavator and Perspex/acrylic sheet.