#1 The series resistor topology slowly builds your peaks in the source, charging the cap until the max is reached. That max will stay charged in the cap. If you get a strong peak, it's full level may not be reached immediately, you may require a few repetitious hits at peak voltage for the cap to achieve the top voltage. This may filter out some occasional high frequency interference. If the interference is repetitious enough, that charge will eventually make it into the cap.
#2 The resistor in parallel on the cap design finds the peak voltage immediately, (assuming your source signal has no series resistance and has infinite current and the diode is a perfect switch), but, this peak voltage across the cap will slowly begin to discharge that voltage once the peak signal has been taken away. For a large resistor value, this may be very slow. For a smaller resistor value, this may be tuned to a speed good for an audio level meter showing peak power, but still be responsive to bass drum notes.
For #1, you can also use 2 normal silicon diodes back to back in series with your source, with a pull-up resistor in between and get the exact peak voltage, as if you were using a perfect diode with a 0v drop. This method is good for finding the peak voltage of signals right down to a few 50-200mV. Using a pnp/npn transistor in emitter-follower mode, with 1 cap and 2 resistors, would work even better. I typically use 2N3906/2N3904. This buffers your source signal and allows you to simultaneously combing #1 and #2. The transistors are being used as a super sharp amplifying diodes. Good for measuring peaks down below 25mv and your source's series resistance and current becomes much less of an issue since it is being buffered.
With all the above methods, you must make sure your source signal is not AC coupled. IE: Have a series cap in the signal line like audio line level/headphone signals. There must be a DC path for those methods to work.