We use IAS to fly by.
CAS is an intermediate step.
EAS is used for structural calculations.
TAS is used for aerodynamic calculations.
G/S is used for navigational calculations.
If there were no system or gauge errors IAS = CAS. Ultimately IAS is at one end, G/S is at the other - you can correct in either direcction as required.
Your term HAS is IAS, because it's a first go at CAS from measured Pt and Ps. CAS is IAS (your HAS) with the system errors calibrated out. In most cases, we act as if IAS=CAS; not strictly true, but so long as the certification process ensured that operating speeds are correct in IAS, close enough.
I'm with Lomcevac here - EAS is essential for engineering calculations, but I really can't see why it's of any value to a pilot operating an aeroplane.
G