Sometimes simply called what it is "De-rated Thrust"
The terminology has varied a bit over the years .. however, one needs to be a bit careful here ... think, and plug in whatever particular words you prefer, the following ...
(a) "de-rate" functionally is the same as changing the installed engine on the wing. That is to say, the entire certification basis is reworked. Main risk for pilots is that this usually involves lower Vmcg/Vmca values for the lower rated takeoffs .. if you have the physical ability to push up the throttles, then you could run into handling problems in appropriate circumstances if you were to do so in the event of a failure
(b) "flex" functionally is using part throttle for the takeoff
(c) clearly you can have a derated takeoff and then apply a flex reduction on top of the derate
Qantas, back in Wal Stack's days as performance boss, was one of the original instigators of reduced thrust takeoff performance .. I can recall Wal, as a part time lecturer when I was a BE student at Sydney, telling us of the early experiments when the captain thought it a tad odd to see the FE PUSH the throttles UP on the 707 when he called for climb thrust to be set .. Wal then introduced a requirement that min reduced thrust was to be limited by the climb thrust setting to avoid such eye-catching things occurring.