I dug a well without a cover before, and someone else's cow fell and died. Am I responsible?

Yes, but not exactly. Because the cause of this thing is that you didn't cover it properly, which led to the death of someone else's cow. It belongs to safety measures that are not in place. If someone falls, it is your full responsibility, and you have not set up safety signs and corresponding safety guidelines. Pay attention to these aspects next time. Fortunately, it is not human! Don't need too much cover!