They are correct, it is the first full service airport in 30 years - I some how don't think you will see your My Travel A330 going to Barbados or a 767 to New York leaving LCY any time soon! - Robin Hood can handle just about anything.
The name on the departure boards, tickets and the one soon to be appearing in brochures will be 'Doncaster Sheffield'.
Robin Hood is the brand, in the same way John Lennon is the brand for Liverpool.