A cumulant is defined via the cumulant generating function
$$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants have some nice properties, including additivity- that for statistically independent variables $X$ and $Y$ we have
$$
g_{X+Y}(t)=g_X(t)+g_Y(t)
$$
Additionally, in a multivariate setting, cumulants go to zero when variables are statistically independent, and so generalize correlation  somehow. They are related to moments by Moebius inversion. They are a standard feature in undergraduate probability courses because they feature in a simple proof of the Central Limit Theorem (see for example here).
So the cumulants are given by a formula and have a list of good properties. A cumulant is clearly a fundamental concept, but I'm having difficulty figuring out what it is actually measuring, and how it is more than just a computational convenience.

Question: What are cumulants actually measuring? What is their conceptual meaning? Are they measuring connectivity or cohesion of something?

I apologise that this question is surely completely elementary. I'm in low dimensional topology, and I'm having difficulty wrapping my head around this elementary concept in probability; Google did not help much. I'm vaguely imagining that perhaps they are some kind of measure of "cohesion" of the probability distribution in some sense, but I have no idea how.

Cumulants have many other names depending on the context (statistics, quantum field theory, statistical mechanics,...): seminvariants, truncated correlation functions, connected correlation functions, Ursell functions...
I would say that the $n$-th cumulant $\langle X_1,\ldots,X_n\rangle^{T}$
of random variables $X_1,\ldots,X_n$ measures the interaction of the variables
which is genuinely of $n$-body type.
By interaction I mean the opposite of independence. Denoting the expectation by $\langle\cdot\rangle$ as in statistical mechanics, independence implies the factorization
$$
\langle X_1\cdots X_n\rangle=\langle X_1\rangle\cdots\langle X_n\rangle\ .
$$
If the variables are jointly Gaussian and centered then for instance
$$
\langle X_1 X_2 X_3 X_4\rangle=\langle X_1 X_2\rangle\langle X_3 X_4\rangle
+\langle X_1 X_3\rangle\langle X_2 X_4\rangle
+\langle X_1 X_4\rangle\langle X_2 X_3\rangle
$$
so the lack of factorization is due to $2$-body interactions: namely the absence of factorization
for $\langle X_i X_j\rangle$.
The $4$-th cumulant for variables with vanishing moments of odd order would be
the difference $LHS-RHS$ for the previous equation. Thus it would measure
the "interaction" between the four variables which is due to their conspiring all together
instead of being a consequence of conspiring in groups of two at a time.
For higher cumulants, the idea is the same.
Cumulant are definitely related to connectedness. For instance for variables whose joint probability density is a multiple of a Gaussian by a factor $\exp(-V)$ where $V$
is quartic, one can at least formally write moments as a sum of Feynman diagrams.
Cumulants are given by similar sums with the additional requirement that these diagrams or graphs must be connected.
Some references:

Chapters 1 and 4 of the book "Path Integrals in Quantum Mechanics" by Zinn-Justin.

My article "Feynman diagrams in algebraic combinatorics". Although it applies to a somewhat different context, it explains the combinatorics of Feynman diagrams in terms of Joyal's theory of species.

The review "Feynman diagrams for pedestrians and mathematicians" by Polyak. However the discussion (e.g. in Section 4.5) of the issue of convergence does not give an accurate idea of this research area.


