When deriving the Bellman equation for $q_\pi(s,a)$, we have
$q_\pi(s,a) = E_\pi[G_t | S_t = s, A_t = a] = E_\pi[R_{t+1} + \gamma G_{t+1} | S_t = s, A_t = a]$ (1)
This is what is confusing me, at this point, for the Bellman equation for $q_\pi(s,a)$, we write $G_{t+1}$ as an expected value, conditioned on $s'$ and $a'$ of the action value function at $s'$, otherwise, there is no recursion with respect to $q_\pi(s,a)$, and therefore no Bellman equation. Namely,
$ = \sum_{a \in A} \pi(a |s) \sum_{s' \in S} \sum_{r \in R} p(s',r|s,a)(r + \gamma E_\pi[G_{t+1}|S_{t+1} = s', A_{t+1} = a'])$ (2)
which introduces the recursion of $q$,
$ = \sum_{a \in A} \pi(a |s) \sum_{s' \in S} \sum_{r \in R} p(s',r|s,a)(r + \gamma q_\pi(s',a'))$ (3)
which should be the Bellman equation for $q_\pi(s,a)$, right?
On the other hand, when connecting $q_\pi(s,a)$ with $v_\pi(s')$, in this answer, I believe this is done
$q_\pi(s,a) = \sum_{a\in A} \pi(a |s) \sum_{s' \in S}\sum_{r \in R} p(s',r|s,a)(r + \gamma E_{\pi}[G_{t+1} | S_{t+1} = s'])$ (4)
$q_\pi(s,a) = \sum_{a\in A} \pi(a |s) \sum_{s' \in S}\sum_{r \in R} p(s',r|s,a)(r + \gamma v_\pi(s'))$ (5)
Is the difference between using the expectation $E_{\pi}[G_{t+1} | S_{t+1} = s', A_{t+1} = a']$ in (3) and the expectation $E_{\pi}[G_{t+1} | S_{t+1} = s']$ in $(4)$ simply the difference in how we choose to express the expected return $G_{t+1}$ at $s'$ in the definition of $q_\pi(s,a)$?
In $3$, we express the total return at $s'$ using the action value function
leading to the recursion and the Bellman equation, and in $4$, the total return is expressed at $s'$ using the value function
leading to $q_\pi(s,a) = q_\pi(s,a,v_\pi(s'))$?