Convention of closed loop gain in op-amp for inverting configuration

I tried to analyze real op-amp equations, in inverting and non-inverting configurations. I found a reference question here in which they were calculating exactly what I wanted to do: calculating the op-amp bandwidth with the same ideal gain and different configurations.

While trying to solve the same problem, I realized that the OP in the linked question and others are doing something may not seem obvious to me, and that was where I've got stuck if were not for that answer.

From the schematic: Basically, the real relation of input/output of the inverting configuration is

$$\ \frac{V_o}{V_i} = -\frac{A_0}{1+\frac{A_0R_{in}}{R_{in}+R_{f}}}(1-\frac{R_{in}}{R_{in}+R_{f}})\$$.

And with a constant GBP (Gain Bandwidth Product), the relation $$\ f_{cl}A_{cl} = f_0 A \$$ gives the new bandwidth. But, if this is more straightforward with the non-inverting configuration, seems that in this case $$\ A = A_0 (1-\frac{R_{in}}{R_{in}+R_{f}}) \$$ which seems o be the attenuating factor. This was the assumption made on the linked question before and seemed generally accepted from the answers.

Now, my misunderstanding is that I was thinking to consider this equality:

$$\ A_{cl} = \frac{A_0 (1-\beta)}{1+\frac{A_0}{\beta}} \$$ and $$\ A = A_0 \$$, while seems accepted to use $$\ A_{cl} = \frac{A_0}{1+\frac{A_0}{\beta}} \$$ and $$\ A = A_0 (1-\beta) \$$ to fit in the equality $$\ f_{cl}A_{cl} = f_0 A \$$, in that case bringing to the correct result of a lower bandwidth. The conclusion that I can make is that the rule is to consider the closed loop gain always as $$\ A_{cl} = \frac{A_0}{1+\frac{A_0}{\beta}} \$$ with A0 the intrinsic gain of the op-amp.

I felt by doubt a bit more legit when I was reading a review of the book "Design With Operational Amplifiers And Analog Integrated Circuits" by Sergio Franco (is it good?), and a comment was: "The author incorrectly refers to the amplifier’s forward path gain “a” as the “open loop gain”. This is wrong because the open loop gain is, in fact, the gain around the opened feedback loop. Indeed, the open loop gain is more commonly referred to as the loop gain. A quick perusal of any feedback control theory textbook should disabuse doubters of this common fallacy.", which seemed my issue here.