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.