Gyroplane is the only term used in the ANO and similar. Autogyro is partly descriptive and not too bad. Gyrocopter is a horrible word conceived out of ignorance and should be avoided at all costs, if not to invite ridicule (sadly, it got used several times in the latest AAIB bulletins).
The suffix -opter derives from pteron meanin wing, so what is a gyroc?
Amateur built are quite common, with PFA overseeing permits in the UK, I believe.
Ken Wallace (the subject of said AAIB report) is one of their greatest proponents, still flying his own designs at the age of 90.
A 2 seater has recently gained UK approval of some sort. I'll leave it to someone more knowledgable than me to expand the detail.