Can anybody clarify the AUTO, AUTO ECO, and A/C climate control modes?

Auto turns on/off the compressor as necessary.
Auto ECO has a more aggressive policy to make sure felt temperature is correct, but using less energy.
A/C - you turn on/off the compressor yourself. I'd never use it unless a situation I cannot predict, requires you to manually turn on/off the compressor.
The user's manual has all the necessary information anyway.
PS: any A/C system in any car built in the last 10 years with automatic climate control (electronic) turns on/off the compressor as required by the set temperature, inside/outside temperature, humidity, light exposure, etc.

 

i think even the gen 3 has variable speed a/c compressor. and even when in manual mode, the a/c runs sometimes, even though the button light is off.

 

As implied above, no, the compressor does not run all the time in any "A/C" mode. In a Prius, "A/C" means literally, "air -conditioning-", not just cooling. It could be heating, or just running the fan, or even nothing operating.
Also, as stated above, "eco" mode tends to run the system at less than full power, and even in "normal" mode "A/C", the compressor, if running, is variable speed. It will be run at whatever power level the system deems necessary. Including zero.

The Prius is a 21st century vehicle, "This is not your grandfathers car."

 

European models have 3 Settings .

Standard,Eco and Fast .
Is that different to the US models ?
In the European owners manual it is said that "Fast" will speed up the fan,I am hoping that setting will defrost the windshield a bit faster .

 

yes, we have nothing like that. i suppose auto eco is similar to your eco, and auto equals standard.
fast is an interesting term, i would like to know the details in the owners manual.

 

um, a/c is related to temp. and auto is related to temp.