Sorry for the confusion. I, unwittingly, used the same notation for the open loop gain of the system and the gain of the opamp. To avoid this confusion again let's use slightly different notation this time.
Ao - open loop gain of the system
Avo - gain of the opamp
β - feedback factor of the system
F=Cf/(Cf+Ci+Cs)
1. Also, let's first establish a common ground in understanding. I assume we can agree that if the gain of the opamp is ∞, then the closed-loop gain of the system is
Ao∞=1/β=-Cs/Cf and consequently the feedback factor β=-Cf/Cs
2. As I said before, the loop-gain is:
To=Avo*[Cf/(Cf+Ci+Cs)]
This follows from circuit analysis if we break the loop at the output of the opamp.
Since by definition To=Ao*β and we know To and β, one can find if necessary Ao from here. I'm not going to do it.
3. Again by definition, for any feedback system
Acl=Ao/(1+fAo)=1/f*[fAo/(1+fAo)] =1/f*[To/(1+To)]=1/f*[1/(1+1/To)]
We already know that To=Avo*F but here Avo is not Ao and F is not β.
It follows that Acl=-(Cs/Cf)*[1/(1+1/FAo)]
4. While a non-inverting opamp configuration directly maps into the general feedback block diagram, the inverting one does not. If you want to map it you need to find the two main parameters, namely Ao and β. But those are not Avo and F. That's why I said not to mix F and β in this case. A shunt-shunt feedback configuration that is driven by a current input signal directly at the inverting input of the amplifier can map directly, but here we drive with a voltage through Ci and things change slightly.
I hope it's clear now.