If you sent the letter in post 9, that frankly would be a big mistake. It's not an apology, they will see it as a wind-up. Now there's isn't much point in arguing that, the damage appears to be done. But the email in post 5, with the "intention to prosecute" is your last-chance-saloon. Forget about what the inspectors on the train told you. They are at the risk of personal violence due to their job, so they will say all sorts of things to de-escalate the problem, since once they have done that it's easier for them to fill in their forms and get the matter handed over to the DRPU. That's life.
So look at post 23. You need to grovel, really grovel, the outlines of the grovelling are indicated to you in
@WesternLancer first few bullets. Check here before sending, but don't hang around too long. If you don't want to grovel, then the court system awaits. That's a tough message but I believe it to be correct.