Contradictory results from power spectral densities and autocorrelation in shot noise
Your Zener diode will have an intrinsic/paracetic capacitance (you might also have connected a separate capacitor in your circuit as well, across zener' output or nearby) and assocaited to it will be a RC time constant with the resistance across it. For the noise to show it's random nature accross this circuit's tapping point, we would have to wait for a time that is  certain order of the time greater than the time constant. That is why a slower sampling rate is able to better capture that.

That is why you observe that when the sampling rate was 1MHz, the autocorrelation was high all accross and when the sampling interval is of the order of Milliseconds, you have very much acheived a good correlation value because you are nearing the typical time constant of an RC circuit, for the half second sampling the noise fluctuations have indeed taken effect across the tappping point as now the time is much greater than the time constant and you observe true randomness, that is short noise that is white.