Agreed that a continental-style naming system works best.
Inter City - WCML, ECML, GW, MML, GA London-Norwich
Regional Express - Cross Country routes, Transpennine routes, SE HS1 routes, London Airport Expresses, etc
Regional - Busier medium distance routes like on Thameslink, Brighton Mailine, etc and longer distance local services like Birmingham - Shrewsbury/Hereford or Manchester-Blackpool
Local - As it says on the tin, short routes on stopping services in busy urban cities. Can be branded into something for each city GBR - Birmingham, GBR - Leeds etc.
For me I would separate Inter-regional out of Regional Express, creating a total of 5 categories (with another one for HS2), with defined service standards:
Intercity Express (ICE): for the services on HS2.
Intercity (IC): basically everything from the current LNER, EMR Intercity, Avanti, all Intercity Express Train services by GWR, and the Voyager services by CrossCountry, plus EMR London - Norwich, and the Scottish Inter7city as well. They should have proper seat reservations, 2+2 seating in Standard Class, 2+1 seating and dining in First Class.
Interregional (IR): long-distance non-intercity services, for example, Waterloo - Weymouth, Waterloo - Exeter, Kings Cross - Kings Lynn, Liverpool - Norwich, Birmingham - Stansted Airport, Liverpool - Newcastle, Euston - Crewe, etc. They should have both standard class and first class.
Regional Express (RE): medium-distance express services, for example, St Pancras - Corby, St Pancras - Dover, Waterloo - Portsmouth, Bedford - Brighton, Cambridge - Brighton, Peterborough - Horsham, Liverpool - Cleethorpes, Birmingham - Liverpool, Cardiff - Southampton, etc. They can further be branded by the region, like Watelroo - Portsmouth can be a South East Express, Liverpool - Cleethorpes can be a Northern Express, Cardiff - Southampton can be a South West Express, etc.
Regional (R): stopping services outside metropolitan areas, for example, Waterloo - Basingstoke, Charing Cross - Dover, Kings Cross - Cambridge stopper, Liverpool - Manchester Airport stopping service. Similar to Regional Express above, they should also be branded by the region.
Suburban (S): local stopping services within a metropolitan area, which should be devolved. They should be operated like the metro with defined line names and consistent stopping patterns. Examples include London Overground, Merseyrail, West Midland Railways, and the future Bee Network. The stops exclusively served by suburban services should even be taken out of the national GBR ticketing and into the local transport ticketing scheme.
The abbreviations are compatible to the wider European network and still make sense in an English context.