The name comes from the FS1 nomenclature for the Shivan Super Laser.
Exactly why I said it makes sense. GTVA beam weaponry is built up on Shivan Beam weapon Theory - they had no Flux Cannons to take apart and study (Unless Inferno canon states that FS1 era Shivans used beams aside from the Lucifer. I forget). Therefore, GTVA beam weapons, while being similar to the Lucifer's Flux Cannons, probably don't actually utilize the same method of operation, and as such don't get the same designation.
Now, the EA on the other hand had, at the very least, the wreckage of the Lucifer's Flux cannons, and I really don't think it's too much of a stretch for one of the Lucifer's arms to have survived the blast, like in the FS1 end cutscene. Therefore, the EA has an actual, semi intact set of Flux weapons to research and back-engineer to create their beams, meaning that their beam weaponry can have the Flux Cannon moniker applied to it.