What should I call a "differential" which cubes, rather than squares, to zero?

I would just call it a module over the truncated polynomial algebra $k[D]/D^3$. Your two flavors of homology appear as positive odd-degree and even-degree groups in $\operatorname{Ext}^*_{k[D]/D^3}(k,M)$. (This is $2$-periodic in positive degrees.) The "exactness" should hold precisely if $M$ is free as such a module.

The same interpretation works also for usual chain complexes, by the way. The algebra appearing there is the exterior algebra $k[D]/D^2$, and the $\operatorname{Ext}$-groups are actually $1$-periodic in positive degrees and recover the usual notion of homology.

If you want to be fancy, you can localize the derived category of $k[D]/D^3$ (or $k[D]/D^2$) by killing free modules. If you do this correctly, you obtain the so-called "stable module category" (a stable $\infty$ or dg-category), in which the mapping complex from $k$ to $M$ is actually a fully periodic version of the above $\operatorname{Ext}$, so in some sense is precisely described by your two different "homologies" of $M$.

Many years ago, when I was a graduate student, I remember seeing a couple of papers on the homologies of operators satisfying $\partial^p=0$, generalizing the case $p=2$. I seem to remember that they were by somebody like Steenrod, and it might evan have been in the Annals, sometime in the 40s or 50s.

Unfortunately, I'm at home now and not able to access MathSciNet to look it up. However, I do remember that there was something like a set of axioms, generalizing the Steenrod axioms, for the various '$p$-homologies' that could be associated to topological spaces using such operators.

I forget now why I was interested in them. When I'm back in my office (maybe tomorrow), I'll try to find it on MathSciNet.

Such objects, for more general values of $3$, or at least the graded version (i.e., $\mathbb{Z}$-graded objects where $D$ is a degree one map with $D^N=0$) have attracted some interest in the representation theory of finite dimensional algebras in recent years, under the name of "$N$-complexes".

The fairly recent paper

Iyama, Osamu; Kato, Kiriko; Miyachi, Jun-Ichi, Derived categories of $N$-complexes, J. Lond. Math. Soc., II. Ser. 96, No. 3, 687-716 (2017). ZBL1409.18013.

may be of interest to you for the fairly lengthy list of relevant references in the introduction.