The shivan flux beam cannons probably were flux cannons. But, the way that gtva reverse engineered them probably forced them to come up with different methods to create a gtva tech beam cannon. What i mean to say is, from the technology the shivans had, the gtva probably couldn't reproduce it 100% so they made a weapon just like it operating on similar principles, just using different stuff and methods to do it. Probably hence the photon beam cannon naming for the gtva beams. Until slang term beam cannon came around.
Gtva beams are related to the lucifers flux beam cannons, but not 100%, they're probably a technology all of their own now that has similarities with the lucifers flux beams.
It should be safe to say that gtva beams are really a different thing than the lucifers, albeit similar in what they both do, but use methods that would be alien to one another.
On the note of shivans and flak cannons, Perhaps V could have made flak cannons for shivans that at least utilized different sound and visual affects. Like the shivans having a rapid plasma spitting gun that had exploded red plasma in proximity to your ship. You know it'd be the same weapon, but at least the shivan flak variant would have different sound affects, projectiles, and explosion. And then shivan flak would look more natively shivan for the shivans, and then we wouldn't have this question of where'd the shivans get flak from, it's just like ours.