I'm not a lawyer; I'm not your lawyer.
In this case, if you don't have a valid ticket (or refuse to produce one), and don't provide your name and address, the officer of a railway company may detain you1 (my emphasis):
If a passenger having failed either to produce, or if requested to deliver up, a ticket showing that his fare is paid, or to pay his fare, refuses or fails on request by an officer or servant of a railway company, to give his name and address, any officer of the company may detain him until he can be conveniently brought before some justice or otherwise discharged by due course of law.
Essentially, if you don't have a ticket and are asked to show one, it would seem that you are required to provide either your name and address, or pay your fare.
The only cases I can find where s 5(2) has been cited were:
- Covington v. Wright  2 QB 469
The defendant boarded a bus and travelled further than the fare was paid for. However, being a case involving a bus and a train, one would assume that the legislation and therefore the case is not authoritative for trains. However, the London Passenger Transport Act cited herein seems to have the same effect as the Regulation of Railways Act.
- Ormiston v. Great Western Railway Company  1 KB 598
This case involves a passenger travelling on a valid, first-class ticket, then being detained by a porter who claimed that they had held only a third class ticket. He was released and was not entitled to special damages simply as a result of slander.
On the relevant question of citizens' arrests, it was found later that2:
Court had to consider the question whether it is necessary to allege that the words impute an indictable offence, and it was held that it was not necessary. Pollock B. in giving judgment said: “The expression ‘indictable offence’ seems to have crept into the text-books, but I think the passages in Comyns' Digest are conclusive to shew that words which impute any criminal offence are actionable per se.
That is, the notion that the offence must be indictable seems not to be based in common law.
As for your question about your rights - the power to arrest you though the punishment is only a fine does not support an action for special damage on its own. You should probably consult a lawyer if you are so detained, as there does not seem to be anything in the Act that allows the company to prohibit you from doing so - or from doing else, for that matter, so long as you do not breach the peace or act in some other tortious manner.
You would also likely to be free to pay the fine (or produce the ticket, if you had one and had not done so by this point in time), or provide your name and address. If you were detained without cause - that is, you were not given the option to supply your name, or not given the opportunity to produce your ticket or pay the balance of the fare, then you may be entitled to damages under a claim of an action of false imprisonment.
Apologies for the lack of links to the cases - I've used paid databases so they likely wouldn't work, but you can try to find these cases in BAILII or some other database you might have access to.
1. Regulation of Railways Act 1889 52 & 53 Vict c 57 s 5(2)
This legislation has had numerous amendments and restrictions applied - you should consider the full extent of these as applicable to the specific location where such an incident occurs.
2. Hellwig v Mitchell  1 KB 609