What is pure energy?
"Pure energy" doesn't mean anything in physics. Energy can take many forms (mass, kinetic energy, or any of many forms of potential energy), but no one of them is "pure" in any sense, no more so than any other form.
That being said, I would guess that when non-physicists use the term "pure energy," they're likely referring to some form of energy that isn't directly associated with a particle of matter. The most common example is probably electromagnetic radiation. For example, as Ted said in his answer to the linked question, when matter and antimatter annihilate, you start with some matter (fermions) and end with no matter (no fermions). But the final state still has energy (since energy is conserved), so in the popular view, one might label the energy of that final state "pure" because the state is devoid of what we commonly consider matter. Since photons (EM radiation) are common reaction products for these processes, they get associated with the phrase "pure energy."
Please note: I am not an astronomer - astronomers, please call me out if I am mistaken.
'Pure energy' is one of those terrible science journalism phrases that usually refers to hard electromagnetic radiation. Energy is a property of light and matter and not a substance in itself. In nuclear fusion, nucleons combine to form bound states with lower total energies than their separate components and the difference is mainly released as high energy photons and neutrinos.
Black holes do not devour themselves when they get too big. A quasar is an active galactic core, where a supermassive black hole at the center of a galaxy is accreting huge amounts of matter. This releases a lot of radiation in two ways - the accretion disk gets very hot and glows in the x-ray region, whilst the characteristic polar flows of a black hole are streams of gas/plasma moving at relativistic speeds which will light up whatever they collide with.
When people say "pure energy", they mean energy which can be readily used to lift a weight off the ground. So photons are "pure energy", because you can absorb photons into work, but massive particles have a rest-mass which requires a process to turn into work, like dumping into a black hole, and so is not readily available.
Pure energy is any field energy, like potential energy, any kinetic energy, like a fast moving particle, but no mass energy of stable or nearly stable massive particles which would require a process to turn into work.