Indeed the notice must contain "an explanation of why the person is being charged a penalty fare" which in this case is "Travel beyond validity". That's a bit of a stretch of the normal meaning of the word "beyond" so definitely worth an appeal on the grounds that it wasn't issued in accordance with the regulations i.e. state that you were travelling between, not beyond, the stations named on your ticket. Then quote whatever you can find (some suggestions already on the thread) to argue that catching the earlier connecting train was a sensible thing to do and didn't render the ticket invalid. If what you did wasn't meant to be valid, then the train companies failed to explain this sufficiently as they are required to do by the NRCoT ("We will make available clear information about the range of Tickets to help you make a well-informed choice about the most appropriate and best value Ticket for your journey.", "When purchasing your Ticket, we will make available information on specific restrictions that apply to your Ticket (for instance the train services on which you can use your Ticket or the route(s) you are entitled to use) and, where possible, any known changes to planned services.", "Some Tickets require you to travel on a specific train service or services, or the services of a particular Train Company or Companies. This will be made clear to you when you purchase your Ticket." - that you believed you could take the earlier train, and the other sources you can quote, could be used to try to demonstrate that they breached those terms. Where did the train company tell you while retailing the ticket that you could take an earlier Northern service but not an earlier TPE service, if that's correct (I haven't checked)? It's also interesting how many people have the opposite problem, buying walk-up tickets and thinking they are only valid on the service they selected. The industry urgently needs to find a way to solve this consistently across all retail channels!)
You've only uploaded part of the penalty fare document so you should check the rest of it against the regulations yourself too. Make sure for example that it tells you somewhere on it that you are entitled to a receipt if you pay the penalty fare - if it doesn't, it similarly doesn't conform to the regulations.
