I think "intercity rail" is a concept that is well-known worldwide and is separate from the former BR InterCity sector. The division between intercity rail and regional/commuter rail, even if not explicitly called that, can be seen everywhere from the US to France to India.
Using the fairly simple and commonly used definition of "intercity rail" as "services focused on connecting major urban areas across longer distances", the operators of intercity rail in the UK are fairly easy to identify - these are the ones I'd call "intercity":
- Avanti West Coast
- Caledonian Sleeper
- Chiltern Railways (London-Birmingham services)
- CrossCountry
- East Midlands Railway (EMR Intercity services)
- Eurostar
- Grand Central
- Greater Anglia (London-Norwich services)
- GWR (former Great Western franchise services)
- Hull Trains
- LNER
- Lumo
- ScotRail (Glasgow/Edinburgh-Aberdeen/Inverness services)
- TransPennine Express