"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."