Consider two systems of spins in an external magnetic field. Suppose system A has all the spins in their high energy state (wrong way given the field) - so it has the most negative temperature possible. Suppose system B is similar, except a few of the spins are in their low energy state - so it has a slightly less extreme negative temperature. If the two systems interact, the few low-energy spins ought to even out between the two systems, which means energy has flowed from A to B (lower temperature to higher temperature).
Your example above, where all spins are in their highest possible energy state, really has temperature T=-0, an infinitesimal temperature just below absolute zero. This would not be the negative temperature possible.
You are correct that energy would flow out of this maximal-energy system. But it corresponds to System B in my example - the system at the less negative temperature.
The animation near the top of the article shows this nicely. The particles approach their max possible state as T flips to negative and approaches zero from below.
Temperatures of positive and negative infinity are statistically equivalent in this example, where states of any energy are equally likely.
It’s only at smaller less-negative temperatures where the population inversion occurs.
Two systems at different temperatures, when put into thermal contact, will come to thermal equilibrium at an intermediary temperature. Total entropy (over both systems) will have increased.
