Well first off, whether or not an internal combustion engine is called a motor or an engine differs by region for English-speakers and in fact in my language there's only "mootor", for both electric and combustion engines.

Secondly, the engine on an enginecycle could also be a two-cycle engine, that's common in small engined dirtbikes and moped particularly.

Thirdly, this may depend on your local legislation, but where I live, a low-power e-bike would be a moped and a high-power e-bike would be a motorcycle. In terms of the license requirements anyway. If it's limited to 25 km/h and requires pedaling, then it's an assisted bicycle and requires no license and no helmet (provided you're over 16).