Non-inverting Lowpass Filter with Capacitor in Feedback
If you are asking why you don't see the second circuit with no R2, that's because it has (with ideal components) infinite DC gain.  At DC the impedance of C1 is infinite and the gain will be (one plus) the ratio of the feedback impedance to R3.

Also, in this non-inverting configuration, the minimum stop-band gain is unity.

In the inverting configuration, you do in fact sometimes see the circuit without the parallel resistor, when integrating low-frequency signals over a finite period of time.   With a short enough period of time, the very high DC gain will not have time to dominate, assuming the DC component is sufficiently small.