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A steam powered cycle would clearly be an engine-cycle, as they are steam engines, not steam motors!
Sticking your arse over a boiler? Not this little black duck!
But then we use gasoline engines, not gasoline motors.
You are correct, sir - and sadly, we are all both currently and historically inaccurate. Then again, back when 'moto-cycles' were first created and named, most people had never seen an electric motor!