Most airline tickets are issued in international booking systems that do not recognize Cyrillic characters — only Latin characters.

This spelling is universal and is accepted by any airline and on any route, even if you are flying a domestic flight within Russia and using a Russian internal passport.

— But I bought a ticket on the airline's website, and everything was in Russian!

Yes, that could be the case if it was a Russian airline. Some of them operate in a Russian (and Russian-language) booking system, so they can issue tickets with names written in Cyrillic. At the same time, the Russian system recognizes both Cyrillic and Latin characters, so if you had issued the ticket with your name in Latin characters that time, you would have used it and completed your flight just the same.

And Kupibilet works with more than 700 airlines, most of which are foreign, selling tickets only through international (English-language) systems. These systems only accept Latin characters, and Cyrillic simply cannot be entered there. 

How to enter the passenger's last name, first name, and patronymic

An error was made in the passenger's last name, first name, or patronymic