Hardening refers to strength gain after final set has occurred. Thus, the ACI definition seems to imply that green concrete has reached final set but hasn't gained much strength. The saw manufacturers' definition implies that some hardening or strength gain has occurred because green sawing should begin when concrete won't spall or ravel at the cut edges. Green concrete can also mean freshly placed (presumably plastic) concrete (John Wiley & Sons Inc).
However, the term is usually applied to concrete that has lost its plasticity. If the term is used but not defined in specifications, you should ask the specifier for a definition. Unfortunately, it probably won't be any more precise than the three definitions described above.
Concrete that uses less in energy in its production and produces less carbon dioxide than normal concrete is green concrete. The other name for green concrete is resource saving structures with reduced environmental impact for e.g. Energy saving, co2 emissions, waste water.