A neat little corollary to this is to look a little more closely at what temperature actually is. "Temperature" doesn't appear too often in the main explanation here, but it's all over the "student's explanation". So... what is it?

The most useful definition of temperature at the microscopic scale is probably this one:
1/T = dS / dU,
which I've simplified because math notation is hard, and because we're not going to need the full baggage here. (The whole thing with the curly-d's and the proper conditions imposed is around if you want it.) Okay, so what does that mean? (Let's not even think about where I dug it up from.)

It's actually pretty simple: it says that the inverse of temperature is equal to the change in entropy over the change in energy. That means that temperature is measuring how much the entropy changes when we add or remove energy. And now we start to see why temperature is everywhere in these energy-entropy equations: it's the link between them! And we see why two things having the same temperature is so important: no entropy will change if energy flows. Or, in the language of the article, energy would not actually spread out any more if it would flow between objects at the same temperature. So there's no flow!

The whole 1/T bit, aside from being inconvenient to calculate with, also suggests a few opportunities to fuzz-test Nature. What happens at T=0, absolute zero? 1/T blows up, so dS/dU should blow up too. And indeed it does: at absolute zero, any amount of energy will cause a massive increase in entropy. So we're good. What about if T -> infinity, so 1/T -> zero? So any additional energy induces no more entropy? Well, that's real too: you see this in certain highly-constrained solid-state systems (probably among others), when certain bands fill. And you do indeed observe the weird behavior of "infinite temperature" when dS/dU is zero. Can you push further? Yes: dS/dU can go negative in those systems, making them "infinitely hot", so hot they overflow temperature itself and reach "negative temperature" (dS/dU < 0 implies absolute T < 0). Entropy actually decreases when you pump energy into these systems!

These sorts of systems usually involve population inversions (which might, correctly, make you think of lasers). For a 2-band system, the "absolute zero" state would have the lower band full and the upper band empty. Adding energy lifts some atoms to the upper band. When the upper and lower band are equally full, that's maximum entropy: infinite temperature. Add a little more energy and the upper band is now more full than the lower: this is the negative temperature regime. And, finally, when everything's in the upper band, that is the exact opposite of absolute zero: the system can absorb no more energy. Its temperature is maximum. What temperature is that? Well, if you look at how we got here and our governing equation, we started at 0, went through normal temperatures +T, reached +infinity, crossed over to -infinity, went through negative temperatures -T, and finally reached... -0. Minus absolute zero!

(Suck on that, IEEE-754 signed zero critics?)

And all that from our definition of temperature: how much entropy will we get by adding a little energy here?

Thermodynamics: it'll hurt your head even more than IEEE-754 debugging.