Some power-on-demand only e-bikes can hardly be confused with, let alone categorised as, bicycles. For example, the Noped is a term used by the Ministry of Transportation of Ontario for e-bikes which do not have pedals or in which the pedals have been removed from their motorised bicycle. These are better categorised as electric mopeds or electric motorcycles.
E-bikes mostly use motors and battery options from a few major suppliers: Bosch, Yamaha, Shimano, and Brose. A few other brands exist, but are less reliable or powerful. Some, like the Yamaha system, have more torque and others are quieter. But generally all four make good options. Look for motor output (in watts) which will give you an idea of total power. But watt hours (Wh) is perhaps a better figure to use—it takes into account battery output and life to give a truer reflection of power.