When did the Roman Empire stop being the Roman Empire

[Robert Vermaat]

That thing in the east was an empire--and occasionally a very good empire, though often a pitiful one--but less and less Roman.

That's again defined by a personal opinion of what you consider to be 'Roman' and what not.

When Rome fell it had not been the capital for one-and-a-half centuries. That's quite some time.
As to religion, was Christianity 'Roman'? Initially it wasn't, but when Rome fell (I trust you refer to 410? Rome was sacked three times during the 5th c.) it had been for the same period.
As to culture and language, nothing changed much in the Eastern Roman Empire. Only after a long time was Latin dropped as the language of the government and the military. But Arrian and Polybios (and a good number of other authors) published their works in Greek, which meant the 'Romans' (including the emperors) could read it.

Etc. etc. etc.

As a linchpin of history, the Gothic sack of Rome is as good a point as any, as is the foundation of Constantinople, accepting Christianity, the Last Western Emperor, the Fall of Byzantium, or Napoleon.