http://geacron.com/home-en/?&sid=GeaCron815987
Oftentimes, in a modern context, they'll refer to them as the Byzantine Empire, and not really the Roman empire. Is this where the identity of Roman was lost? in 629? Or is it a more modern idea that the Byzantines were Greeks, and not the Eastern Roman Empire?
The only event I can think of that would correspond to the "change" in 629 was the fact that Heraclius ceased using Latin titles for himself and exclusively used "Basileus," the Greek title for "Sovereign." This thread, particularly the post by /u/shlin28, tries to explain the significance of this change.
Maybe the interactive map interprets the decision to drop the Latin titles as a turning point where the Eastern Roman Empire permanently adopts a more Greek identity.
Anyway, Byzantine is just a term used by modern scholars to designate the Eastern Roman Empire after semi-arbitrary points such as after the "fall" of Rome in the West or the death of Justinian. The Byzantines always called themselves Romans. Here's a thread that details this very well. Look at /u/BruceTheKillerShark's post. I'm just repeating what's already been said on this sub.