In my (British) youth, the term "roundabout" was used for both the disc with handles which you had to "scoot" (or run!) round, and for the roundabout on the fairground, with horses going up and down, or the smalled version with cars, fire-engines, teddy-bears, mickey-mice etc. - the children's version. I'd be frightened of calling the disc anything other than a turntable, simply BECAUSE the park or recreation-ground roundabout was always fitted with handles/handrails.