You have two dependent variables $a$ and $w$. So, there is a joint distribution $p(w, a)$. You can make a marginalization by one of them, pretty much as you did in your second formula. $$p(w) = \int p(w, a)da$$ $$p(w) = \int p(w | a)p(a)da$$
The only difference in this case, the calculation made for the specific point $x_i, y_i$, which is empathized by sub-index on $p_i$ and conditioning on $D_i$
The key thing is that we can calculate the target distribution in many ways. With likelihood, evidence and prior you could indeed find posterior, but they are not always tractable/available. So, that's why in literature we usually differentiate true posterior and approximate posterior (or just posterior). Usually we get $p(w|d)$ with some form of approximation, but in the paper authors decide to get closed-form solution. That's why it was useful to represent in another way with intermediate distribution $a$. This would allow them to get closed-form for different activation functions.
So it's posterior in the sense of general context, not that specific formula, I think your limits of Bayesian knowledge is just fine