OK, I think I’ve figured out what you’ve done – you’ve effectively used the ‘seasonally-corrected trend’ column of the Mauna Loa data (though I think you’ve taken a longer route to effectively doing this). Each monthly value represents that month’s entry in that column minus the previous month’s. I’ve plotted that and it looks extremely similar to your ‘deviation from mean’ graph.
I’m not sure what this can show, other than perhaps degree of intra-annual autocorrelation. Perhaps I’m not up-to-speed on what Web is claiming?