In answer to your question, the most likely answer is that the LF351 was more expensive so they used a cheaper op-amp where high slew rate was not required. The LM307 is/was a 741 type of op-amp, slow and semi-precision. Slew rate is less than 1/20 that of the LF351.
It's important to have a high slew rate amplifier in the LF351 position because there the output of the input op-amp should move quickly from -0.7V to +0.7V (ideally instantly) since one diode or the other should always be conducting.
To see how it works, consider when the AC-coupled input is > 0, say V1. Current flows through R1 and R2 and D1 so the left side of R3 is at -V1, thus the current flowing into the node at pin 2 of the LM307 is Vin/R4 - Vin/R3 + Vout/R5, so the output voltage (ignoring C2 for the moment) is +Vin
When the AC-coupled input (right side of C1) is < 0, call it -Vin, D2 conducts (keeping the op-amp output from straying too far from ground, reducing the demands on slew rate) and the current flowing into the node at pin 2 of the LM307 is -Vin/R4 + Vout/R5, so Vout = -Vin (again ignoring C2).
C2 acts to average the output voltage with a time constant of C2*R5.
In the second case there is a resistance R2 + R3 between pin 3 of the LM307 and pin 2 of the LF351. Since both nodes are always at virtual ground there is (ideally) no effect from that 200K resistance.