I have recently been bootstrapping the confidence intervals of an NN model estimated to data.

I execute the following psudo-code, which seems similar to previous bootstraps I have done:

 1. Take all observations of data $x_i$, estimate $\hat{y} = \theta(x_i)$
 2. Resample $N$ observations from the set of $x_i$ with replacement. Estimate $\tilde{y} = \tilde{\theta}(x_i)$. Repeat until we have $M$ different versions of $\tilde{y}_m$
 3. Calculate the standard deviation of everything in $\tilde{y}_M$ as $\sigma_{\tilde{y}}$ and then the prediction interval of $\hat{y}$ is $\hat{y} \pm z * \sigma_{\tilde{y}}$ where z is chosen to be an appropriate value from a standard normal distribution, usually 1.96 for 95% CI.

See [here for more reading on bootstrap intervals][1].

At the same time, I see other approaches that use the quantiles of $\tilde{y}_M$ in order to construct intervals of some sort, such as [here][2] and [here][3]. The intuition for such an argument is very strong and seems valid - but is unfamiliar.   What's going on with the use of quantiles? Is something else being calculated (prediction intervals, etc.) that are similar in flavor but are not the same? 


  [1]: https://ieeexplore.ieee.org/abstract/document/5966350
  [2]: https://stats.stackexchange.com/questions/226565/bootstrap-prediction-interval
  [3]: https://online.stat.psu.edu/stat555/node/119/