How exactly are the phase shift and cutoff frequency defined for a filter + amplifier circuit?
To be clear and to protect against future edits, here is the circuit you show that this answer pertains to:



There is no low pass filter here.  That is because the negative input of the opamp stays at 0 V during normal operation.  The voltage across C2 doesn't change, so there is no current thru it, so there is no difference if it were removed.

With C2 removed, this is simple block with a gain of -1.  It does have frequency limits over which it works, but those are not well defined.  One limit is a result of the gain-bandwidth product of the opamp.  However, that is usually only specified as a minimum value, with little guidance how high it might be.  There are other low pass filters due to parasitic capacitance, but again, those are hard to quantify.

If you want a predictable low pass filter with a single dominant pole, then you need to add your own deliberate filter component, like a inductor or capacitor, that dominates stray and parasitic values.