What is the prime sign that a country is transitioning from a developing to a developed economy?
I don't know what the 'first sign that a country is transitioning from a developing to a developed economy', but I would look at the rates of output growth and study its acceleration. 

In a layman's words, a transition from low to high income regime requires convergence to a high income teady state. This is only possible if the growth rate of aggregate output, underpinned by improvements in all aspects of the economy, has been consistently higher than that observed in high income (which you call developed) economies. Transition is complete (or the country achieves a 'developed' status) when the economy's growth rate reaches its long run steady state rate (typically low (&lt;2% p.a.) positive growth rate).