Dry atmospheric temperature profile (dry adiabatic vs. isothermal)
The two articles you found are spot-on. An isothermal atmosphere is indeed the condition that maximizes entropy for a given amount of energy. Yet a positive lapse rate is almost always observed. Temperature typically decreases with increasing altitude. The key to resolving this apparent paradox is how heat flows into and through the Earth's atmosphere. The Earth's atmosphere is a system that is very far from equilibrium.

If the atmosphere isn't in that maximum entropy state (and it never is, at least not from very bottom to very top), heat transfer will set up to put move it toward that state. Very key: That entropy-driven heat transfer rate is rather low. If other heat transfer processes are in play (and they always are), it is very easy to overwhelm that low entropy-driven heat transfer that would nominally equalize the temperature. The Earth's troposphere is heated from below and cooled from above. This alone overwhelms the entropy-driven heat transfer, and by multiple orders of magnitude.

The other factor is how heat flows through the atmosphere. The presence of greenhouse gases in the atmosphere set up the conditions that make a positive lapse rate possible. By way of analogy, imagine a perfect blackbody heat source that generates heat at a rate $Q$ and has a surface area A. If that heat source radiates to the dark sky, the equilibrium temperature will be given by the Stefan-Boltzmann law, $T_\text{top} = (Q/(\sigma A))^{1/4}$. (Note: I'm assuming that $Q$ is much, much greater than the tiny amount of heat coming in from the cosmic microwave background.) If you put a perfect blackbody blanket atop that heat source, the blanket will radiate at a rate $Q$ outward and inward. The heat source will receive heat from above and below, making the equilibrium temperature of the heat source with one blanket equal to $(2Q/(\sigma A))^{1/4}$, or $2^{1/4} T_\text{top}$. Yet another blanket makes the thermal equilibrium equal to $3^{1/4} T_\text{top}$, and so on.

The atmosphere with its greenhouse gases that are optically thick in the thermal infrared acts as a bunch of blankets. Obviously not perfect blackbody blankets, but blankets nonetheless. This is what favors the development of a positive lapse rate.

This favoring isn't universal. There are times when weather creates conditions where the lapse rate is negative, that is, temperature increases with altitude. This turns out to be a very stable condition (the name for this is a "very stable atmosphere"). This negative lapse rate precludes rising and falling parcels of air. With calm air as well, the only thing that can transfer heat is diffusion, which is a very slow process in the atmosphere.