Erm, too long. You need to keep things concise and stop rambling. Is there really any need to reiterate every bit of what happened on the day, as the RPI would have submitted an account of that anyway?
Keep to your main points. You were of the understanding that tickets could be bought on the train but now realised that your obligation is to buy tickets before boarding if possible. There was no opportunity on the train. When you went to pay the RPI on the bridge, you were mistaken as showing your Railcard to get through. Attach your old tickets if you wish, and apologise for the whole event.