The confusion may be as a result of the fact that The melodic Minor means different things in different tradiations.

In Classical music, the melodic minor is considered to have an ascending part and a descending part.

The Ascending part is the same as the major scale but has a flattened 3rd.

The descending part has a flattened 3rd but also has a flattened 6th and a flatttened 7th.

The descending part is therefore the same as the Aeolian mode, potherwise known as the natural minor.

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Confusion often arises out of the fact that in Jazz the melodic minor is the same in ascent as it is in descent, i.i, it is the same as a major scale with a flat 3rd going up and down the scale.

Jazz distinguishes the melodic minor as a seperate resource to the natural minor/aeolian mode.

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I don't know the tune, but I suspect that it is probably meant to be in the Dorian mode as that and the myolydian were generally the resources from which melodies were derived in western folk music.

The Dorian mode is the same as the major scale with a flat 3 and a flat 7.