Let $\{b_n(\omega)\}_{n=1}^\infty$ be a sequence of random variables in the underlying probability space $(\Omega,\mathcal F,\mathbb P)$. Let $b\in\mathbb R$.
$b_n\xrightarrow{p} b\equiv\forall\epsilon>0\;\;\mathbb \lim_{n\rightarrow\infty}\mathbb P(\omega\in\Omega:|b_n(\omega)-b|>\epsilon)=1$
The way I understand this definition is the following: given $\epsilon$, for each $n$, $\mathbb P(\omega\in\Omega:|b_n(\omega)-b|>\epsilon)$ is a sequence of real numbers, say $a_n$, so $\lim_{n\rightarrow\infty}\mathbb P(\omega\in\Omega:|b_n(\omega)-b|>\epsilon)=1$ refers to the limit of that sequence of real numbers: $\lim_{n\rightarrow\infty}a_n=1$.
What happens if each $b_n(\omega)$ comes from a different probability space? For example:$(\Omega_n,\mathcal F_n,\mathbb P_n)$. Then we would have:
$b_n\xrightarrow{p} b\equiv\forall\epsilon>0\;\;\mathbb \lim_{n\rightarrow\infty}\mathbb P_n(\omega_n\in\Omega_n:|b_n(\omega_n)-b|>\epsilon)=1$
Is this correct? Is this necessary? Or the first definition embodies this second case.
You have it exactly right. The random variables don't need to be defined on the same space, because $b$ is a constant. OTOH if $b$ were a random variable then you would need all the $b_n$ to be defined on the same space in order for the quantity $b_n(\omega)-b(\omega)$ to make sense.