I think it must be clear that the FADEC is not a part, it is a SYSTEM.
Let me quote again the AMM:
General
The FULL AUTHORITY DIGITAL ENGINE CONTROL (FADEC) provides full range of engine control to achieve steady state and transient engine performances when operated in combination with aircraft subsystems. The FADEC System consists of a dual channel Electronic Control Unit (ECU) and the following peripherals:
- Hydromechanical Unit (HMU)
- Dedicated Permanent Magnet Alternator (PMA)
- Variable Stator Vane (VSV) and Variable Bleed Valve (VBV)
- High Pressure Turbine Clearance (HPTC) control valve
- Low Pressure Turbine Clearance (LPTC) control valve
- Rotor Active Clearance (RAC) control Valve
- Starter Air Valve (SAV) and ignition components
- Thrust Reverser control
- Fuel return valve control
- Engine Sensors
- Electrical harness
- Burner Staging Valve (BSV)
- Engine configuration plug.
The ECU is a vibration isolated single unit mounted on the fan case, no
forced air cooling. It is a part of the basic engine equipment.
So "Engine" for the E makes more sense than "Electronical"