When to read Battery voltage and battery current during charging
Maybe you are getting confused as the PWM of the both systems are for different functions. The PWM of the Microchip document controls the buck converter transistor. There are 2 PWMs in the NEC document one controls the buck converter transistor, but the other controls a charge control transistor. The latter is used for the voltage and current measurement decision.

So in the case of the NEC document it is like pjc50 mentions, there is no current flow when the PWM is off (for the charge control transistor), so you cannot measure the current there. It has some advantages of measuring the voltage of the battery when no charging (or discharging) current is applied, as you are closer to the real open circuit voltage of the battery. Only closer because of the relaxation effect of batteries, which is on a seconds to minutes timescale, so much slower than your typical PWM signals.

Why exactly it would result in erroneous operation if you would measure the voltage and current during the on time of the PIC PWM is not really obvious to me. The only hint I could find was that the PWM gets disabled and adjusted after the measurements, which should be done when PWM is low (otherwise you will get strange PWM pulses).

As the PIC solution involves a buck converter but does not use the charge control transistors, there will always be a current flowing to the battery regardless of the state of the PWM, so you won't have a benefit of getting closer to the open circuit voltage.

Generally you want to get your measurements as close to the open circuit voltage as possible if you are doing voltage based state of charge indication. So ideally you want to measure the voltage if no current is going in or out of the battery and have waited for some minutes to let the battery settle (relaxation effect), waiting is usually omitted. The current would introduce an error because of the internal resistance, so the voltage you measure would be too high while charging and you'd estimate the state of charge as too high.

To be on the safe side, you'd still switch from constant current to constant voltage mode when the measured voltage reaches 4.2 V, primarily because you don't monitor the internal resistance of the battery at the same time to calculate the internal cell voltage. (And I think that approach to fast charging was recently patented for whatever reason)