In terms of mechanical differences, the InterCity 125/175/225 were all based on locomotives moving a train of unpowered and shuntable carriages; the IEP/IET trains are multiple units with a mix of powered and unpowered carriages in fixed formation.
To answer the naming question: it's all in the capitals, and all so simple...
- The Inter-City: a named train of British Railways introduced in 1950 from Paddington to Wolverhampton via Birmingham;
- Inter-City, later InterCity: British Rail's brand for high quality, long-distance trains, and later the name for the sector which ran those trains;
- intercity or inter-city (no capitals): a generic term for high quality long-distance trains, derived from British Rail's brand but now used widely around the world, including branding in Germany (IC) and France (Intercités), though it has no formal definition or common standard;
- Inter-City 125/InterCity 125: British Rail's enduring public brand name for two Class 47 locomotives sandwiching 7 to 9 mk3 carriages; similar branding used for the InterCity 225 and (briefly) Intercity 175;
- High Speed Train/HST (with capitals): the generic, non-branded, classification given to the InterCity 125;
- high speed train (no capitals, not 'HST'): any train that travels at high speed;
- Intercity Express Programme (IEP): the project to develop a new fleet of high speed trains, leading to the development of the Class 800 and 801 trains from the Hitachi AT300 family, originally proposed as the Hitachi Super Express Train;
- Intercity Express Train (IET): though it sounds generic, only officially used as Great Western Railway's brand name for the Classes 800/0, 800/3 from the IEP and for the almost identical 802/0 and 802/1 trains ordered separately, which are all 'bi-mode' trains capable of running on diesel or electric;
- Azuma: exclusively LNER's brand name for bi-mode trains in Classes 800/1 and 800/2, and electric-only trains in Classes 801/1 and 801/2, all ordered under the IEP.
So all IEP-derived trains are high speed trains (no capitals), but none are High Speed Trains (with capitals); both the IET and InterCity 125 are high speed, intercity (no capitals) trains, but only the latter is a High Speed Train; whilst only the High Speed Train and GWR's 801s and 803s are InterCity/Intercity (with capitals).
I think.
Clear?