How I understand the Box-Jenkins Method in a nut-shell is that a time-series model has signals that can be identified by weighting its own past lagged values, or weighting its owned past errors or some combination of the two processes. If sequential lags are significant, you are identifying trends in the model. If periodic lags are significant you are identifying cycles (seasonality) in the model. None of these patterns can be adequately identified statistically until you are sure that your data is not integrated (test for stationarity). If your time-series achieve stationarity and you have identified all possible significant lags sequential and periodic (seasonal), what is left will be white noise (no signals). Contrast this approach to Triple-Exponential Smoothing or unobserved components model forecasting. At least to me, it is easy to see (visualize) how a time series can be decomposed to a trend, seasonal, cycle, and an irregular component. You can then easily forecast out the clear signals (trend, seasonal, etc...) and then you can make assumptions about the irregular component in order to forecast out future values of that more noisier component, combined the pieces in order to have an m-step ahead forecast.
I find, with the box-Jenkins approach, I cannot visualize easily how it isolates trend and seasonal components nor how it combines these pieces to produce m-step ahead forecast as I can with the other structural time-series approaches. Too me using correlograms to talk about how to identify the order of the model is not the same as helping to understand how the pieces work together to produce a forecast, as you can show with a decomposition method. Are there more intuitive (visual) ways to explain what is going on with ARIMA models or sources I can refer to get a different view of this methodology? For instance, what can I expect my graph of a time-series to look like if it is ARMA(2,0) vs ARMA(2,1) vs ARMA(2,2)? Is there some logical progression or pattern in the graph of the data that makes intuitive sense?