It is surely as simple as the fact that the person in question did not travel on the railway at any time on the date that the train company is claiming. That is an absolute defence to the allegation and therefore no fine is warranted or justifiable.
Hence, thank you and good day? In the general spirit of any legal dealings, answer only the question that is asked.
That an unusual journey took place on a different day is not at issue. But in the context of that journey, the travel was legitimate (due to the non-working ticket machine) and an explanation was given to the on-train staff, who themselves only took the details and said that they would "possibly" get a letter or a fine "but probably not". As such they are surely entitled to rely on the advice given by the railway staff, and their own knowledge that they have done nothing wrong, so therefore no need to suddenly start pleading guilty to incorrect allegations of fare-dodging that occured on dates and times when they were not present on the railway, and definitely no need to pay any fines that were not even issued.
Would welcome other thoughts on this, but I would have thought that a plain denial is all that is required here. No need to complicate the matter with other anecdotes about travel on other days, unless and until an allegation is made about that particular day's travel.