"I am with the concept that the harmonic minor (take Am for example) is just C major started in a different place"

The name of this type of minor is the "natural minor". Also known as the Aeolian mode.

The "harmonic minor" is the same as the "natural minor", but with a sharpened 7th note.

The "melodic minor" is the same as the "natural minor" but with both a sharpened 6th and 7th note.

The difference between a major and a minor scale comes down to one thing only - the 3rd note of the scale.

If it has a minor 3rd, it is minor. If it has a major 3rd it is major.

So another way of looking at the melodic minor is to see it as a major scale with a flat 3rd.