The OPAMP is powered by +12 and -12V (minus is weak as it is created by simple charge pump converter). The zener D6 at the opamp output is OK there, otherwise the voltage may overshoot to +/-12V and destroy the ADC.
The RV1 for amplitude regulation is wrong. You have to use an voltage divider for that (but you do not need as the voltage is limited by zener D6).
The RV1 there is just a part of the timing constant RV1C10. Unless you set RV1 so low that the OPAMP will not be able to feed enough current to the RV1.
Mind the output currents of the Opapm could be 20mA max, so double check.
The D3=1n4148 diode at the opamp output works as a simple peak detector with time constant ~RV1C10.
Based on the first schematics, the max peaks ampl at ADC input will be -0.6 to 3V, with time constant of 0.5usecs (1/2piRV1C10). So the pulse will disappear with aprox this time when the opamp output goes to zero. Not sure you would be fast enough to sample such short pulses properly.
Now - what you want actually want to measure??, as it seems to me you are not clear about that..
If you want to sample the "shape" of the pulses you need an ADC with sample rate at least 10x the 1/(pulse_duration).
If you want to detect the peak amplitude of each pulse you need a longer time constant - similar to the 1/frequency of your pulses into the coil and a good timing between shooting the pulse into the coil and sampling that pulse at the output.
If you want to measure the average amplitude of a train of pulses then you may use large timing constant (ie 10-100ms) and measure the average voltage with the ADC without any need for fast adc.
So what you actually want to do? Mind the electronics is about good understanding of the circuitry and using MATH
PS: I see the frequency of your pulses is 650Hz, 70us/1400us.
So the RV1 shall be for example ~10kohm and C10=10-22nF. That gives you aprox 500-1400us time constant and together with the diode D3 it will work as a simple "peak detector". You would have about 0.1 x timeconstant to sample the peak till the voltage drops significantly at the ADC input (so something like 50-100us). The precise timing depends on the shape of the pulse, however. So an oscilloscope may help.
Again - you have to specify WHAT you want to measure at the output of opamp to decide on proper signal processing approach..
PS: adding the 1k in series with front of the ADC input will limit the clamp current to the lower clamp diode inside the chip to aprox I = -0.6V/1kohm, the clamp diodes are usually rated for I<<5mA.