Can cumulants arise from another property of the distribution.?

Question

Recall that cumulants are defined as \begin{align} k_n = \frac{d^n}{d t^n}\log (M_X(t)) |_{t=0}, n \in \mathbb{N}, \end{align} where $M_X(t)$ is the moment generating function of a random variable $X$.

Question: I am curious if cumulants can arise from some other property of the distribution $P_X$.
For example, moments can be shown to arrive from three different 'properties' of the distribution of $X$:

1. From the distribution: $E[X^k]= \int X^k(\omega) dP(\omega)$
2. From the CDF: for $X\ge0$ $E[X^k]= k \int_0^\infty t^{k-1} (1-F(t)) dt$
3. From characteristic function or moment generating function $ E[X^k]= \frac{d^n}{d t^n}M_X(t) |_{t=0}$

In this spirit, I am interested in whether there are alternative ways of obtaining cumulants.

One possible such way its to use the connection between cumulants and Bell polynomials, which essentially relates $\frac{d^n}{d t^n}\log (M_X(t)) |_{t=0}$ to $\frac{d^n}{d t^n}M_X(t) |_{t=0}$.