From the wikipedia page for the CFM56-engine: notice that the different thrust ratings are specified per aircraft type, with for instance the 737-700 having been delivered with four different ratings. The engine hardware for a model number does not change, a reduced maximum rating is indeed just a different circuit board inside the FADEC.
The smaller versions have a lower thrust rating because of the shorter moment arm between vertical tail and engines: upon engine fail, the rudder authority is less and can therefore only compensate for a lower max. thrust.
This article mentions the various ratings, thrust limits, derating etc that are applicable to fixed wing and rotary wing aircraft. Notice the use of the word "rating":
The only 2 thrust ratings that airplane jet engines are certified for are maximum takeoff (MTO) and maximum continuous thrust (MCT). All other thrust ratings —such as(ground/flight) idle, climb and cruise thrust— are only figures recommended or set by the engine manufacturer.