well... both.
Nitrogen is the element that we are talking about, "N" for short.
N is a sticky element, it usually has some oxygen "O" or hydrogen "H" attached to it. An N with 3 H's stuck to it "NH3" is ammonia, NO2 is nitrate, etc etc.
living creatures need N to survive - that would be your
eating or sometimes they want the O or H that's stuck to it and the N is leftover - that's your
producing. Like when you cycle your tank you are growing bacteria that swap O's for H's to turn ammonia into nitrite.
But things aren't "producing" or "eating" the N in the sense that they are adding or removing it from your tank. They are just sticking it to something so that it hides from your test, until another critter comes along and unsticks it. The N is in your tank because it is part of fish food, it only leaves the tank if you take out an algae or bacteria or poop or whatever it is stuck to, or bacteria turn it into a gas [N2O i think, not sure].
The thing with cyano is that it is really really good at finding the N it needs. So it can unstick it from lots of things. When people talk about cyano making its own nitrate, they are talking about "nitrogen fixing" which is another way of sticking N to stuff. That makes it special and different from, say, green hair algae which is pickier about what the N is stuck too and can't "fix" its own. So there's not much point in trying to get rid of cyano by just lowering nitrates. it's too clever for that, and will just get N someplace else
tl;dr - we don't usually talk about the whole nitrogen cycle, and the part that's missing is cyano. In this pic you can see how fixing (yellow) fits with the nitrifying (purple) and denitrifying (green) bacteria that make up our whole cycle, and completes the circle.
Cyano does have weaknesses. It needs light, phosphates, and not too much current. So you can play around with those if your tank is out of balance such that there is too much cyano and it's ugly. There are also a bajillion types of cyano, so some tricks work better on some kinds than others.