Apologies - one final question- why did he read my rights?
If the inspector intended to make a report of a matter that might proceed to prosecution and was satisfied that an offence was evident, it would be right for him / her to caution you in accordance with the Police & Criminal Evidence Act (1984) codes of practice, before questioning. (The 'caution' is what many people erroneously refer to as 'read me my rights', which is American jargon, but probably understandable because it's so often heard & seen on TV and in films)
Sorry if it sounds worse than it is, but I always think it's best to know exactly where you stand. Failing to co-operate by not providing your details when asked will have been the trigger for calling for Police assistance
From your description of the event there are three charges that they may now consider. The first two are strict liability Byelaw offences, the simplistic descriptions are:
1. Breach of National Railway Byelaw 19 (2005) - may be summonsed for remaining in and travelled in First Class when holding a Standard Class ticket only.
2. Failed to give name and address when requested by rail staff in connection with a breach of Byelaw 19. - may be summonsed as an offence contrary National Railway Byelaw 23 (2005)
3. The third possibility is more serious, if the inspector reports that s/he believed your intention was to travel in First class with only a Standard class ticket with the intention of avoiding payment of the correct fare unless challenged, an offence contrary to Section 5(3) of the Regulation of Railways Act (1889) might be summonsed.
As others have pointed out, it is best to be brief but factual in your reply now you have received a letter from the TOC.
it may be that recognition of your liability in this and a careful apology and offer to make recompense will achieve a settlement without Court action, but any decision on that lies squarely with the TOC.