This paper considers the impact of Lyman α coupling and X-ray heating on the 21-cm brightness-temperature one-point statistics (as predicted by seminumerical simulations). The X-ray production efficiency is varied over four orders of magnitude and the hardness of the X-ray spectrum is varied from that predicted for high-mass X-ray binaries, to the softer spectrum expected from the hot interstellar medium. We find peaks in the redshift evolution of both the variance and skewness associated with the efficiency of X-ray production. The amplitude of the variance is also sensitive to the hardness of the X-ray spectral energy distribution. We find that the relative timing of the coupling and heating phases can be inferred from the redshift extent of a plateau that connects a peak in the variance's evolution associated with Lyman α coupling to the heating peak. Importantly, we find that late X-ray heating would seriously hamper our ability to constrain reionization with the variance. Late X-ray heating also qualitatively alters the evolution of the skewness, providing a clean way to constrain such models. If foregrounds can be removed, we find that LOFAR, MWA and PAPER could constrain reionization and late X-ray heating models with the variance. We find that HERA and SKA (phase 1) will be able to constrain both reionization and heating by measuring the variance using foreground-avoidance techniques. If foregrounds can be removed they will also be able to constrain the nature of Lyman α coupling.