Why are numerator coefficients named $b$ and denominator coefficients named $a$?
My suspicion is that this ordering comes from the difference equation, which in most texts precedes the $\mathcal Z$-transform:

$$a_0 y[n] + a_1 y[n-1] + \cdots = b_0 x[n] + b_1 x[n-1] + \cdots$$