Blackbird - Richard Thompson and Lennon/McCartney.
Country Life - Trad (Watersons) and Steve Knightley

It's also worth mentioning that many traditional songs are known by more than one title. Some people prefer to use the first line of the song, some use the first line of the chorus, others pick up a phrase from somewhere else in the song... and there's a song called 'As I Roved Out' which doesn't have that phrase in it anywhere!

Even contemporary songs suffer - Pete St. John's 'Rare Ould Times' is often referred to as 'Ring a Ring a Rosie' (from the first line of the chorus), which confuses it with a nursery rhyme!

I'd say the 'correct' title of a version of trad song depends a lot upon who collected it, and what their source called it, but a contemporary song should keep the title the writer gave it!