OK, here I am after an hour or so of research. Lili Marlene (also Lili Marleen)was written by Hans Leip as a poem in 1915. Apparently he had two friends named Lili and Marleen, and it was their names which inspired the title. It was published in a book of poetry in 1937 and was set to music by the composer Norbert Schultze in 1938. Cabaret singer Lale Andersen started singing the song soon after, and recorded it on 2 Aug 39. It made little impression until 1941, when a copy of her recording began to get airplay on a radio station for German soldiers. It was an immediate hit among the soldiers of many nations, and reportedly has since been translated into 48 languages. The Fassbinder film, Lili Marlene in 1980 is a highly fictionalized account of Lale Anderson's life, quite loosely based on her biography from the accounts I read.

Despite looking reasonably diligently, I was unable to come up with a date for Marlene Dietrich's recording, but I think you would have to assume that it was no earlier than 1941. I don't think there is any question as to the fact that her version was better known and much more frequently played in the United States, and perhaps most of the Allied countries as well, though that is just my assumption.

One thing is certain, without Lale Andersen, there almost certainly would be no Marlene Dietrich version. As to which is better or "more" definitive, I guess that is just a matter of opinion.