Running machinery and fingers, jewellery, bits of clothing, long hair, rags etc just do not mix.
You don't need to fear machines but God help you if you don't respect them. The human body is way more fragile than your average lathe/milling machine/motorbike chain/engines. You get something caught while the thing is running and you run the risk of causing injuries like those above.
Everyone makes mistakes that's for sure but you need to mitigate your risks and be focused on the job at hand at all times when working on/with machinery with exposed moving parts.
Just be careful and pay attention and be patient. Also, the chap with the bike chain didnt really know what he was doing. He should have asked for help or sought advice on how best to clean his chain before doing it "his way" which cost him his fingers. The biggest mistake he made was running the freaking engine. Why not just do it with the engine off, bike in neutral and use your hand to control the turn of the sprocket and chain???