The same website where you link the document has another publication, where they state that the "Global Deviance is -2*max(log likelihood) under hypotheses H0 and H1 respectively." I ran a quick test on my own data, and it seems to match that claim: I ran a simple negative binomial model using gamlss, and it produced a Global Deviance of 6697.049. I then used logLik() to get the log-likelihood of -3348.524. If you do the calculation, it works.
Since you want to maximize the log likelihood when comparing models, and GD is the negative of that, it stands to reason that you want to minimize the Global Deviance output from gamlss (which is intuitive, since, as you note, we normally want to minimize 'deviance'). I don't think the negative GD by itself has a specific meaning. However, from iteration 1 to iteration 11 in your post, your GD is decreasing, which would seem to be a good sign.