People can sue for anything, that does not mean their case has merit. Likewise, many companies choose to settle cases even if they are in the 'right' to avoid publicity or a sympathetic jury setting a precedent. The law is filled with grey areas like "reasonable precautions". I think with millions of people riding POTC and this being the first known incident of fingers being amputated, that most layers would be able to argue that Disney has taken all reasonable precautions. There is also likely video of the actual event that could indicate unsafe actions by the guest. The law does not require companies to protect all people from their own stupidity, but only take those precautions to minimize risk associated with reasonable behavior. In civil cases, juries can assign comparative negligence and split liability (i.e. the person is 75% at fault, but Disney is 25%). So the question becomes how negligent is the behavior of the guest If they were warned to keep their hands in the boat and they knowing did not comply.
In you dog bites child example, If you post a sign but have the dog on a long rope and children can walk within striking distance, yes you open yourself up to liability because you were aware of a dangerous situation but did not take reasonable precautions. However, if you have a 6 foot high security fence with a posted sign and a teenager enters your back yard, then you likely will not be liable because you took reasonable precautions and provided a fair warning. The dog, however, may still be destroyed for attacking a human.