Byzantium is the empire and Constantinople is its capital city.
“Istanbul” is how the Turks heard the phrase, “eis ten Poli”, “in the city”. It is not even a proper name, just a geographical fiction.
Ah, no. Byzantium is the Latin version of the Greek Byzantion, a city named after King Byzas. Constantine remaned the city Nova Roma, but it was popularly called Constantinople. You are however correct about the etymology of Istanbul.
Today "The City" refers only to Manhattan ~