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To expand on that it was during the enlightenment in the early renaissance where people had a boner over the Roman Empire but still thought the medieval Roman Empire (Byzantine) to not be cool. So they came up with a new name for it. A declining empire that had a massive beauracracy, spoke Greek and had the wrong brand of Christianity (Orthodox) is not nice enough to create a glorious image like the Pax Romana did.

This of course made a lot of people upset in the then Ottoman empire since they identified as Romans but were not counted as Romans according to western people. Think "You're not Romans with a glorious history, you're Byzantines" even though they clearly were.

For extra fun the Byzantine/Roman distinction is also unfair.

• Eastern Rome always spoke Greek, even at 200AD.
• Orthodox and Catholic were the same pre-schism.
• During the decline of the Western Empire the capital was moving a lot anyway so "based in Rome" was soon outdated.
• During the decline Italy was just another province anyway so "based in Italy" was soon outdated.
• They were literally the same thing except one half managed to fuck their shit up while getting invaded by hordes of tribes at the same time.