But this is only for distinguishable particles. If you have a bunch of indistinguishable particles, you get the Fermi-Dirac distribution or the Bose-Einstein distribution, depending on whether they are fermions or bosons.
You can find all of these distributions on Wikipedia.
The temperature will go from infinity to -infinity and as you keep adding energy you will approach zero temperature from the left (increasing temperature). The zero value is reached when the energy of the system can no longer be increased and all the atoms are in the E2 state.
A system at finite negative temperature is actually considered "hotter" than a normal system at any positive finite temperature; if you put them in thermal contact, heat will flow in the direction that increases entropy, which you get by taking energy out of the negative-temperature inverted system and adding it into the ordinary positive-temperature system. This increases the entropy of both systems.
The definition where temperature is the "average kinetic energy of the particles" is a special case, and it only really works when that energy is evenly distributed over all degrees of freedom. For example, you wouldn't consider an icy comet to be at a high temperature just because it's moving quickly, even though its particles have a great deal of kinetic energy on average!
The first thing to understand is microstates, which is just the number of ways a system can have a certain energy. Eg in a two-state system with ten particles and energies +/-(E/2), there's one microstate where the energy is -5E (all negative), ten states with -4E (one elevated), etc. Then entropy is just the log of the # of microstates, which is much easier to deal with, since microstates tend to behave exponentially. Eg entropy(E=-10) is log(1)=0, entropy(E=-9) is log(10), etc.
Then temperature like you said is d(entropy) / d(energy). Two systems with different temps brought into contact will exchange energy until the temps are equal, since this configuration maximizes entropy.
The two-state system can have negative temperatures since entropy starts decreasing with energy once more than half of them are in the higher-energy state. This can't happen in more familiar scenarios (eg ideal gas, blackbody, etc) since usually entropy always goes up with energy.
I do understand the benefit of not having 10 different names for the same concept under different scenarios, however. Even if that name isnt the best.
But note that, even in the physics scenario, temperature isnt really the name providing the most intuition either. You need to be aware of the connection between higher temperature -> higher excitability/mobility to begin with, for it to make sense; and temperature isnt the only way to modify this underlying excitability in the first place.