The three qualities of minor scale appear to have arisen in the 19th century as a device to reconcile the chromatic alteration that was prevalent in minor-key harmonies with the new fashion for playing scales as a way of learning to play an instrument. Harmonic minor, on the other hand, is a scale, and a rather artificial one at that. It could also be the name of a mode, such as Dorian or Mixolydian. The first is the 'home' pitch, called the 'tonic,' such as D or A flat, and the second is a quality designation, which is usually either major or minor.
The preposition 'in' denotes the key of a piece. I would summarize the other answers by calling out a faulty premise of the question: there is no such thing as a song 'in harmonic minor.'
