beta of inverting opamp amplifier
The circuit is not exactly equivalent to the two blocks. Some steps must be taken before arriving to that formula.



\$\beta\$ in fact is not \$R_{in}/R_{f}\$. 

It is \$R_{in}/(R_{in}+R_f)\$ instead. Which is the same as the non inverting configuration.

Removing the internal summing node (notice that the signs are brought to the external one!) you get:



(Then if you want you can bring the first block inside the loop, but \$A_{OL}(s) \cdot \beta\$ does not change. And that's the term you'll be considering to analyze the stability of the closed loop system).

If \$A_{OL} = \infty\$, then you get:

$$ A_{CL}=-\frac {R_{f}}{R_{in}+R_{f}} \cdot \beta^{-1} = - \frac {R_{f}}{R_{in}}  $$

In general (not in this case, assuming an ideal OA ), you could also have another block (transfer function) at the output.