A question about the Commutativity Property in discrete signals block diagrams
If you apply the multiplication with $b_0$ at the output, then that multiplier is after the feedback loop. Consequently, if the output (after the multiplier) is $y[n]$ then the input to the delay element is $y[n]/b_0$. The corresponding difference equation is

$$y[n]=b_0\left(x[n]-\frac{a_1}{b_0}y[n-1]\right)=b_0x[n]-a_1y[n-1]\tag{1}$$

which is identical to the one with the multiplier at the input, as expected.