A train derailing can be a metaphor for many things. In an article by Rob England, it is a metaphor for service management as well as problem and risk improvement. The train at Cherry Valley derailed because police did not communicate well to the train to warn of upcoming danger. While the US National Transport Safety Board was doing risk management and driving service improvement after the derailment, while the people cleaning up the aftermath were conducting problem management. There is a lesson to be learned here about service improvement:
Arguably, fixing procedures which were broken was also problem management. The local dispatcher failed to tell the train crew of a severe weather warning as he was supposed to do, which would have required the crew to slow down and watch out. So training and prompts could be considered problem management. But somewhere there is a line where problem management ends and improvement begins, in particular what ITIL calls continual service improvement or CSI. In the Cherry Valley incident, the police and railroad could have communicated better with each other. Was the procedure broken? No, it was just not as effective as it could be.
It is not to say that every improvement is problem management. England emphasizes this by suggesting that improving communication is not problem management because it should have been done with or without the presence of a problem. England notes that a distinction should also be made between problems and risks. Problems are a subset of the much broader heading of risk management, according to England.
Furthermore, to put things more into perspective, England notes that problem management is a part of risk management, which is a part of service improvement. In regards to the Cherry Hill example, service improvement would mean notifying trains more quickly to impending danger. Your organization might not deal with things like train derailment and grave physical risks, but maybe they do. Either way, making the distinction between various areas of problems, risks, and services will allow for you organization to work more smoothly.