There was no real solid concrete moment, but the point that I would define as the relative watershed moment was the reign of Heraclius.

After his campaigns against the Persians, Heraclius adopted the title "Basilius," which is Greek for "King" or "Sovereign," instead of the full and lengthy title that the Emperors usually used. Now, why did he do this? For a variety of reasons. The traditional account is that he simply wanted to reflect the Greek domination of the Empire, but emperors after Heraclius continued to use Flavius Augustus and Caesar in their name's and titles. The theory I subscribe to is that he did so out of sheer convenience. In the life of a military campaigner, you're constantly dealing with dignitaries and men of all kinds of stature. It simply was less difficult (and less intimidating!) for people to just address you as "Basilus," as opposed to the longer title. Alongside this, it made the emperor, in a way, more approachable for the regular soldiers that Heraclius so heavily depended on during his rule.