The temperature is achieved by slamming two nuclei together. There is enough time for the various particles inside to scatter off each other many times and establish a thermal distribution of energies between themselves, but not enough time for any of them to interact with anything outside the two nuclei. After the collision takes place, theses particles will stream out of the interaction region just like in normal proton-on-proton collisions, going through decays and physically traveling through various detector elements.

Presumably, the temperature is inferred by measuring the various energies of the outcoming particles (just like in a normal collision) and noting that they are distributed thermally (i.e. according to Bose-Einstein or Fermi-Dirac statistics). This distribution is tell-tale evidence that the particles exchanged energy through scattering many times.

I guess you could call this thermal imagining, although "thermal imaging camera" denotes something that looks at infrared radiation and measures its distribution (in order to infer the temperature of the matter that radiated it). In this case, the particles themselves (and their decay products) are exploding outwards and having their energies measured.

They handle the heat by having an inconceivably small amount of matter at that "temperature". They measure it by looking at the resulting particles and working out how much energy the collision had, and converting between units of energy and temperature.

There are some problems with describing the energy of a particle accelerator as a "temperature", but it makes the article more accessible and understandable, which is surely a good thing.

I don't think there's much of a problem with calling it a temperature. They don't use that term when describing merely one particle scattering off another, they only use it when they are colliding whole nuclei and there are enough particles to observe a thermal distribution. The key part about having a temperature is to have a distribution of modes (e.g. free particle momentum states) and that there is enough mixing around between these modes to generate a thermal distribution.

Should we be making shitty news articles more accessible or understandable, or should we be making the public more literate and intelligent?

At a certain point, dumbing stuff down strips it of all meaning. Our species is going to have to start taking collective responsibility for its collective intelligence if we want a high tech future.