Answer:
In this context, the modern term "engine" generally means some machine or device which turns energy stored in one form into mechanical motion.
As such, we generally do not consider a device an "engine" if it is powered by human or animal.
However, waterwheels generally can be considered the first type of engine; a water wheel converts the potential energy stored in moving water into rotating mechanical motion. The first known waterwheels were used by the Ancient Greeks (likely somewhere around 500-300 B.C.).