I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer.

The mixed Poisson distribution and compound Poisson distribution are two conceptually different ways to combine the discrete Poisson distribution with PMF $$f_{\lambda_0}(k) := \mathrm{Pr}(X_{\lambda_0}=k) = \frac{e^{-\lambda_0} \lambda_0^k}{k!}, \qquad \lambda_0 \in \mathbb{R}^+,\ k \in \mathbb{N} \tag{1}$$ with another probability distribution to get a new discrete PMF.

In a mixed Poisson distribution, the rate parameter $\lambda_0$ is generalized to a positive-real-valued continuous random variable $L$ with PDF $\pi(\lambda),\ \lambda\geq 0$, so that $f_{\lambda_0}(k)$ becomes the more general PMF $$f_L(k) = \frac{1}{k!} \int_0^\infty e^{-\lambda} \lambda^k \pi(\lambda)\, d\lambda. \tag{2}$$

For a compound Poisson distribution, we take a Poisson-distributed discrete random variable $N_{\lambda_0}$ and an arbitrary discrete random variable $X$ (whose sample space is the natural numbers). We take a countably infinite number of variables $X_i$ that are all identically distributed with $X$ and independent of each other and with $N$, and we form the sum $$Y_{\lambda_0,X} = \sum_{i=1}^{N_{\lambda_0}} X_i. \tag{3}$$ Then the PMF $f_{\lambda_0,X}(k)$ of $Y_{\lambda_0,X}$ is a compound Poisson distribution.

A natural question is whether these families of PMF $f_L(k)$ and $f_{\lambda_0,X}(k)$ are distinct. Although I haven't been able to find any explicit statements about this question, I believe that I can prove that any mixed Poisson distribution whose random rate parameter $L$ is supported on only two positive numbers cannot be a compound Poisson distribution (see my math SE answer for the argument). Assuming my argument is correct, then not every mixed Poisson distribution is a compound Poisson distribution. What about the converse - is every compound Poisson distribution a mixed Poisson distribution? (I propose a somewhat heuristic construction for re-expressing any compound Poisson distribution as a mixed Poisson distribution in my math SE question, but I'm not sure if it always works.)

We can naturally generalize this question beyond individual discrete random variables. The PMFs for mixed and compound Poisson distributions are PMFs for two different classes of Markovian jump processes (i.e. continuous-time and discrete-space stochastic processes) at a fixed point in time: a mixed Poisson process and a compound Poisson process, respectively. Is every compound Poisson process a mixed Poisson process?