I think that the key to understanding these methods and their differences is this statement from the cited JMP documentation: "However, unless you have a very large amount of data, a nonparametric fit might not accurately reflect behavior in the tails of the distribution."
So this situation is similar to reliability analysis. Most of the time in an analysis, the model is used for interpolation, but in reliability and capability, the emphasis is on extrapolation into the tails. The tails of the distribution are where the desired information resides. The parametric fit is more efficient (more information from less data). It is possible (i.e., sometimes) to fit a good parametric model (good estimates of the parameters) with only a handful of observations. This approach emphasizes the importance of your choice for the model and your belief in it.
For example, I fit several distribution models to the weight data in Big Class using the Life Distribution platform for illustration.
I chose the Normal scale to linearize the CDF in this case. The models are very similar over the range of probability from 0.3 to 0.9. They are practically indistinguishable in this range and all of them provide similar estimates by interpolation. What about the tails, though? Calculating the capability requires estimating the weight range of ±3σ, which on the normal scale corresponds to probabilities (1.0 - 0.9973) / 2 = 0.00135 and 1 - 0.00135 = 0.99865. These different models will give very different extrapolations of weight at those probabilities.
The symmetric normal distribution equates the mean and the median. Only the median is guaranteed to represent the center in a non-normal distribution. So the Percentile method is a direct translation of the capability indices.
The Z Score method instead translates the specifications to quantiles but this transformation will still depend on the choice of the model and extrapolation. The specification is first converted to a probability with the fitted distribution model. Then the probability is converted to a quantile from the standard normal distribution, so it can be compared to ±3.