RAT 5
Best source of information on this is the Boeing TakeOff Safety Training Aid or Boeing's FlightOps course (I assume Airbus do something similar)
Need to be careful with definitions here. Here are mine (i'm not saying that they are the global definitions or exhausative, just mine) with some thought. Hope this adds to the discussion.
Derate - 'buttons' on flight deck to reduce thrust as required for an individual take-off to reduce maintenance costs, to lower V1 when VMCG limited (low weight or slippery runways). Flight Manual has separate TO charts as they are certificated ratings.
Assumed Temp/Flex TO - flight deck option to reduce TO N1 or EPR to that seen at a higher temperature for an individual take-off. Used to reduce maintenance costs, particularly when Derate not available or was not purchased. No benefit can be taken for reduced V1 as the pilot may opt to go for full TO rating at any point.
Re-rate - Not a flight deck option. Engine is re-rated to a lower thrust on a more permanent basis by changes to the FADEC. Engines may be rerated to a lower thrust setting when engine temperature margins are becoming small at the higher rating to extend on-wing life. It can also further reduce maintenance costs as the maximum thrust available is limited and the derates or ATM have a lower starting point - lower thrust can be used at low weight operations where max derate/ATM is being used with higher rating.