The mixed Poisson distribution and compound Poisson distribution are two conceptually different ways to combine the 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 an unbounded 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.
I have three very closely related questions:
Q1.
$f_L(k)$ and $f_{\lambda_0,X}(k)$, where $L$, $\lambda_0$, and $X$ are arbitrary, are both families of discrete PMFs. Are either of these families contained within the other? (They clearly aren't disjoint, because if $L \equiv \lambda_0$ and $X \equiv 1$ are deterministic, then $f_L(k) = f_{\lambda_0,X}(k)$ are both the usual Poisson distribution $f_{\lambda_0}(k)$ given in (1).) This question is related to, but distinct from, this one.
I think the answer is that the families are the same, but I'm not sure. If the random variable $L$ has a moment-generating function (MGF) $M_L(z)$, and we let $G_X(z) := E \left[ z^X \right]$ be the probability generating function (PFG) for $X$, then the PGF corresponding to the mixture distribution (2) is $$G_L(z) = M_L(z-1),$$ and the PGF for the random variable $Y$ in (3) is $$G_Y(z) = \exp[\lambda_0 (G_X(z) - 1)].$$ By equating these, we can naively argue that any compound Poisson distribution with a fixed rate parameter $\lambda_0$ and a discrete RV $X$ equals a mixed Poisson distribution with a continuous RV $L$ whose MGF equals $$\exp[\lambda_0 (G_X(z+1) - 1)].$$ Conversely, we can argue that any mixed Poisson distribution whose random rate parameter $L$ has MGF $M_L(z)$ equals a compound Poisson distribution for a positive rate parameter $\lambda_0$ and a discrete RV $X$ whose PGF is $$G_X(z) = 1 + \frac{1}{\lambda_0}\log[M_L(z-1)].$$ (Although I believe that the RHS is only well-defined if we can place the branch cut for the log function such that the function $M_L(z)$ avoids that branch cut on the unit disk centered at $z = 1$, so this construction may not always work.) Of course, both of these arguments are pretty fast and loose, and they may not be airtight.
Q2.
Is there a unique natural way to assign the PMF (2) to a discrete random variable that acts on the same probability space as $Y_{\lambda_0,X}$, so that we can meaningfully talk about whether the random variables are equal almost surely, and not just equal in distribution? If so, is every mixed Poisson random variable equal almost surely to $Y_{\lambda_0,X}$ for some $\lambda_0, X$? What about vice versa?
Q3.
A compound Poisson distribution summarizes the statistics of a compound Poisson process, which is a type of continuous-time, discrete-state stochastic process (i.e. a jump process). I believe that we can also naturally think of a mixed Poisson distribution as summarizing the statistics of a mixture of Poisson processes across different values of the rate parameter $\lambda$. If this is correct, then are either of the family of mixed Poisson processes and the family of compound Poisson processes contained within the other?
I think that Puri and Goldie (1979) at https://www.jstor.org/stable/3213382 gives a partial answer to Q1: not every mixed Poisson distribution is a compound Poisson distribution. Puri and Goldie give the counterexample of a mixed Poisson distribution whose rate parameter $L$ is a binary random variable that takes on only two values $\lambda$. They claim that A. Tortrat (1969) proves that such a mixed Poisson distribution cannot be infinitely divisible. But any compound Poisson distribution is infinite divisible. Therefore, such a mixed Poisson distribution cannot be a compound Poisson distribution. But I haven't been able to find any results about the truth of the converse statement that every compound Poisson distribution is a mixed Poisson distribution.