I've already explained why it has evidential value. The use of the word "proves" is not especially useful when one is talking in the context of defending a notional prosecution, given the low bar for a successful defence (needing to merely cast reasonable doubt upon guilt).
With respect, the OP is not a fare dodger, even if their actions bear an unfortunate resemblance to that kind of behaviour. They have paid their fare, and someone who has paid their fare is, by the very definition of the phrase, not a fare dodger.
Given that the last 40 plus years of my life have been spent working in this very area and for well over 20 years in Courts across the land. I think what the OP really needs to be helped with is understanding how these things often pan out in practice.
Let's say that the Court is presented with something like this:
1. There is no dispute that Mr X travelled,
2. Mr X made to leave the railway at Cardiff and he could not produce a ticket when he was asked to do so
3. Mr X stated he had lost his ticket, but did not produce any receipt or confirmation of any payment at the time of interview
3. When he was given an opportunity to pay the fare due for the journey he had just completed Mr X stated he had travelled from Bridgend and offered to pay from there
4. Because a revenue protection exercise was in operation Mr X was questioned further under caution, which he understood and at that time he then admitted he had in fact travelled for a longer journey from Neath
5. The fare from Neath to Cardiff is considerably more than that from Bridgend to Cardiff (the actual fares would of course be quoted in Court.)
6. The TOC allege Mr X intended to avoid the fare due at the time of questioning.
7. Mr X may be the fifteenth similar case presented by the same prosecutor that morning, all arising from a special revenue check at Cardiff because a major event was bringing many thousand of people into the city and all of whom offered a short fare when challenged by inspectors
The prosecutor may refer to case precedents, which may or may not be wholly relevant, but may have some similar matters of interest. (i.e; Corbyn [1978], Bremme [1965])
Mr X might plead 'not guilty' and might seek to rely on creating 'doubt' as suggested by 'The Love of', but in my honest, practical experience this is very rarely successful in circumstances like this. The prosecutor will press the question 'If you did not intend to avoid the correct fare, why did you not tell the truth about where you travelled from?' Answering 'I was annoyed and wanted to save money' in front of the Magistrates isn't going to help much.
Put yourself in the position of a Lay Magistrate and see how you think you might act, yes, you will be assisted by the Courts Legal Advisor, but LA's do also advise on the practicalities too.
Whilst all cases should be assessed purely on evidence, I'm not daft enough to think that Magistrates will not consider probabilities in their discussions and that's why sometimes a result may seem perverse and why it is right that there is an Appeal to the Crown Court procedure too.
I have to say that I am with 'tiptoptaff' on this one.