What does Adam Smith mean when he says wages are determined by: "advancing, stationary, or declining state of the society."
Taken literary the passages do not hold up, but the passages also have to be read in the context they were written. An important caveat is that Adam Smith actually did not propose any explicit theory of wage growth rather the above quotes are observations he made at the time of the writing (at least I think that the passages when read in the context of the chapter should be interpreted as empirical observations). Take the first quote:


  It is not the actual greatness of national wealth, but its continual increase, which occasions a rise in the wages of labour. It is not, accordingly, in the richest countries, but in the most thriving, or in those which are growing rich the fastest, that the wages of labour are highest.


This observation was made on the basis of him comparing England which was traditionally a backwater - a poor country, with stagnant but 'rich' countries such as Qing dynasty China.  But the observation itself was incorrect. At the time no published real GDP statistics existed and so Smith just thought that China is richer due to the fact that in the West China was often mystified and its wealth was exaggerated. 

As the Maddison's estimates show (see graph at the bottom), at the time of publishing of the Wealth of Nations the Britain was already much richer than China.  However, even at that time it was well known China was stagnant and UK was growing rapidly due to its early industrialization and so Smith conflated those.    

Moreover, also in recent history we can see that Euro-area (comprised of mostly Rich developed countries) in last decade experienced very slow growth - that although technically not stagnation comes close to it - has higher real wages than China which  experiences rapid growth (between 2011 and 2019 it was always higher than $6\%$), yet real wages are higher in the Euro-zone than in China.  

However,  wages tend to raise with economic growth. In fact from macroeconomic perspective there is no meaningful distinction between output and income. Moreover, until very recently average wages tracked productivity very closely and even though in US it is no longer true that they do in many places they still continue to do so. Over long periods of time eventually the output and wages will be higher in countries which are 'thriving' in sense they experience economic growth. This can be shown in the same estimates from the Maddison which shows that although all of the countries in the graph were roughly equally poor in 1500, the countries that experienced high economic growth, are the ones that now have relatively high average wages and standards of living. 

So to sum up Adam Smith was not wrong in observing that wages tend to be higher in the 'most thriving' countries, but he was not correct when he said the wealth of the country - at least understood as the output of the country or GDP is not accompanied with high wages.  

Of course, converse applies as well. A country which experiences negative economic growth will see its real wages and standards of living falling, and stagnant countries will experience stagnation also with respect to real wages.