> there is no actual line between the use of gears and the use of motors.
There's a very clear line in this case. Cycling is a human powered sport. Battery, gasoline, nuclear, and other power sources are not allowed. This is why mopeds are not allowed. Also prohibited is temporary power storage like a capacitor or spring, which might be used for regenerative braking.
You can put all the motors you want on your bike (IMHO) so long as you don't energize them.
Only a specific set of technology is allowed in a bike race. Under UCI rules, cyclist aren't allowed to use aeroshells, and recumbent bicycles are also prohibited, even though both are much more "similar" to a bicycle than your proposed monkey with a walking stick.
The rules attempt to describe the ineffable concept of what it means to be a "bicycle race". They are meant to encapsulate a mass consensus based on tradition and expectations, including the idea of how to balance "human" and "technology". The rules must also face the challenge of changes in technology, and ever-present rules lawyering. The Lugano charter, mentioned earlier, tries to capture and influence that mass consensus.
There are different interpretations of what a "bicycle race" means, which is why there are different rules organizations. If you want to run a race which allows a monkey with a walking stick to compete, feel free!