Understanding this peak detector circuit
Assuming ideal diodes, for \$V_{in} \lt 0\$ this is an inverting op amp with gain \$\frac{R_3}{R_1}\$, and \$R_2\$ keeps the non inverting input from floating. For   \$V_{in} \gt 0\$, this is a buffer.  The circuit probably makes the most "sense" if \$R_3=R_1\$, but if \$R_3 \ll R_1\$, than you are getting something akin to half wave rectification instead of full wave.  

You are correct about the low pass filter, but since you are low-pass filtering a rectified signal, this is more like an rms-filter or envelope detector.

I wouldn't call it a "peak detector", as those usually store the peak value on a cap with no discharge path.  Here, the cap discharges through a resistor, so the voltage is not stored.