Let $X$ be a Polish space and let $\mathcal{M}_1(X)$ be the space of Borel probability measures on $X$ with weak topology. To be exact, the topology on $\mathcal{M}_1(X)$ is generated by the integration maps $\pi_f : \mathcal{M}_1(X) \to \mathbb{R}$ defined by $$\pi_f (\mu) := \int_X \mu(dx) \ f(x) ,$$ where $f \in C_b (X)$ the space of bounded continuous functions on $X$.
Let $(\Omega, \mathcal{B}(\Omega), \mathbb{P})$ be a probability triple such that $\Omega$ is a Polish space. A map $\mu : \Omega \times \mathcal{B}(X) \to [0,1]$ is a said to be a random Borel probability measure on $X$ if for every fixed $\omega \in \Omega$, the map $B \mapsto \mu(\omega, B)$ is a Borel probability measure on $X$, and if for every fixed $B \in \mathcal{B}(X)$, the map $\omega \mapsto \mu(\omega, B)$ is a measurable random variable.
With some slight abuse of notation, we can now define $\mu : \Omega \to \mathcal{M}_1(X)$ as the map $\omega \mapsto \left( B \mapsto \mu(\omega, B)\right)$. This is just to emphasize the random variable notation. Now, since there is a topology on $\mathcal{M}_1(X)$, we can define convergence of a sequence of random probability measures $\mu_n : \Omega \to \mathcal{M}_1(X)$ by stating that $\mu_n \to \mu$ weakly iff $$ \mathbb{E} f(\mu_n) \to \mathbb{E} f(\mu)$$ for any $f \in C_b(\mathcal{M}_1(X))$.
First question: is the above form of weak convergence of random probability measures equivalent to showing that for any $g \in C_b (X)$, and all $f \in C_b (\mathbb{R})$, we have $$ \mathbb{E} f \left( \pi_g (\mu_n)\right) \to \mathbb{E} f (\pi_g (\mu)) .$$ I would refer to this as weak convergence of integration maps as random variables.
In addition to the above form of convergence, I have also encountered convergence of the random variable defined by $\omega \mapsto (\omega, \mu(\omega)) \in \Omega \times \mathcal{M}_1(X)$. Again, we can equip the space $\Omega \times \mathcal{M}_1(X)$ with the product topology, and there is again a notion of weak convergence of such random variables by stating that $(\cdot, \mu_n (\cdot)) \to (\cdot, \mu(\cdot))$ weakly iff $$ \int_{\Omega} \mathbb{P}(d \omega) \ f(\omega, \mu_n(\omega)) \to \int_{\Omega} \mathbb{P}(d \omega) \ f(\omega, \mu(\omega)) $$ for all $f \in C_b(\Omega \times \mathcal{M}_1(X))$.
Second question: what is the relationship between these two forms of convergence of random measures? It is clear that the marginals of the second form of convergence correspond to individually to regular weak convergence of random variables and the first mode of convergence of random measures. But which is the more commonly used one?
The second form of convergence seems considerably harder to prove.