Should I be trying to replicate the NEAS results?

[NEAS: This is an independent student project. The NEAS postings are suggestions to get you started. You can construct an ARIMA model, compare two or more models for a given era, compare two or three eras by type of model of by model parameters, examine seasonality, test out-of-sample forecasts, and many other items.]

It all makes sense now.  The Box-Pierce test first appears in Chapter 16 in the context of just sample autocorrelation (of the series).  It then reappears in Chapter 18 in the context of residual's sample autocorrelations.  So, I am with you that during the first-round white noise test, we can use both Box-Pierce & Bartlett's tests; then once we start fitting, we can use all three tests to see if the fitted model is a good one.