"precision" because it starts rectifying (reproducing input on the output) from nearly zero to -V supply (for the inverting config, while for the one Zap proposed it works from 0 to +V supply. A simpler rectifier made by only diodes will start conducting from 0.6-0.7 V, so it is more distant from an ideal rectifier. You use the "super diode" when you want to get the precise "absolute value" or envelope of a signal, or if you need to fully exploit its amplitude (as in this case, I think).
The one I proposed (which is typical) allows you to either choose to precisely rectify both halfwaves with 0.5 gain (by using 50k,100k,150k as in the scheme) or have an increased gain on the negative halfwave only (by using higher R5/R6 ratio).
Of course, if you cannot use smaller caps, that's a problem, but I'd try with Zap's circuit with a cap on the load, there input impedance is high (+ input of the opamp) and peak detector resistance can almost what you want...
It could be over engineered... I was trying to simplify ("clean") tuning, since in this way you can decouple the various effects. I don't think there's anything wrong with that. Once you get something working, you can even simplify it... An opamp IC costs some cents (yes, LM358 will probably be too slow while a comparator could oscillate), and contains two of them, where one can be used for rectifying, the other for Schmitt comparator...