I have advised twice already in this thread that when you receive a letter from the Company about the incident then you should : check it to see if they have accepted your explanation that your "abbreviated name" is not a false name, and that if they do not, then and only then, you would contact a Criminal Defence lawyer.
It appears that the Company has sent you that letter and they did not accept your explanation. But instead of contacting a lawyer you are writing your own letter and asking us for help with that. It's okay that you don't want to ask a lawyer, but it's difficult giving advice to people who don't follow it. I will give you my advice again in the hope that you take it this time.
If what you have told us on here about your abbreviated name is exactly true and is the whole truth, and if you can provide documentation to demonstrate that you regularly use your abbreviated name in official correspondence, then: Write back to the Company explaining that you did not give a false name and provide them with your evidence including copies of correpondence to you at your current address in that name.
If that is true, then there is no "criminal liability" and there is nothing to settle out of court, just the unpaid Penalty Fare.
If all of that is not all true, then you really should contact a Criminal Defence Lawyer.
If you can write a letter which explains all that and also makes an offer to pay the Penalty Fare which is still unpaid, then what else is there to say? You could even enclose a cheque for the Penalty Fare with your reply.
If you don't feel confident in writing that letter, then you really should ask a professional to do it for you - even the Citizen's Advice bureau might be able to help if its just to explain about your name.
Does it say under what legislation they are considering prosecuting under?
I'm assuming that they will be considering S.5.3 (c) of the
Regulation of Railways Act:
5.3 If any person:
(c) Having failed to pay his fare, gives in reply to a request by an officer of a railway company a false name or address,