It might be too long for a comment so I write it as an answer. Apologies if I missed any ongoing discussions. Let me say the general background first and then my thoughts about OP's question.
The model $y=a+bx+e$ as such tells us nothing about what the true $b$ is. We need more restrictions to identify $b$. There can be many ways of making identifying restrictions. Some are:
$E(e|x)=0$,
$median(e|x)=0$,
$e|x \sim N(0, \sigma^2)$,
the distribution of $e$ conditional on $x$ is independent of $x$,
$E(e|z_1, z_2)=0$ for some other given $z_1$ and $z_2$,
or many possible others. We have 3 => 1, 4 => 1, 3 => 2, and 4 => 2. Each identifying restriction is associated with the "true" $b$ parameter. The true $b$ identified by 1 is not necessary equal to that by 2. We know that the true $b$ by 1 (the probability limit of OLS) may be different from the true $b$ by 5 (the probability limit of 2SLS).
Identification by condition 1 gives OP's "interpretation b". OP's "interpretation a" seems related with option 3 or 4 above, but it is unclear since OP does not give enough information about the population distribution. The distribution of $y$ (or equivalently of $e$) given $x$ should be specified somehow to achieve the identification of the true $b$.
How to interpret $b$ is actually the same as how to define the true $b$, and defining $b$ requires restrictions. Like I said at the beginning, just writing $y=a+bx+e$ is uninformative about what the true $b$ is.
Now to the question, OP says, "the error term fulfills all the OLS assumptions", which I understand as $E(e|x)=0$. Then OP's interpretation "b" is correct. Note that "b" is correct because OP assumed so ("the OLS assumptions fulfilled") not because of any other fancy reasons.
The true $b$ associated with OP's "a" (if the probability distribution was specified) may or may not be equal to the true $b$ associated with $E(e|x)=0$. But if the distribution of $e$ conditional on $x$ is independent of the $x$ value, then its conditional mean should also be independent of $x$. So "a" is also correct if "a" is rephrased to something like: (c) "Given knowledge of the value $x$ takes one would describe their belief about the value $y$ takes with the probability distribution describe by the right hand side under the assumption that the distribution of $e$ conditional on $x$ is independent of $x$". That's because 4 implies 1. Here correct means the true parameter defined by "c" is the same as the true parameter defined by "b".
