The abrsm little red book actually (for once) hits the nail on the head about the notes of the harmonic scale. It points out that the 6th and the 7th in the harmonic minor scale are separated by an uncomfortably large step, but both notes are necessary to build the harmonies of the appropriate minor key, and therefore it is quite necessary to include them in the series of notes that we call the harmonic minor scale.
Otherwise, the whole situation is a lot less complicated than the nomenclature would suggest. There are a lot of notes. A good tune contains only a subset. That's the key. Occasionally extra notes creep in for special effect. They're accidentals. Some accidentals are particularly useful, in order to make music more singable, or in order to fill out chords nicely. The 6th and 7th in pieces with a minor key are particularly prone to alteration.
Many musicians believe that it's easiest to understand the subsets of notes by playing them in order. That's scales. They have a problem with the notes that very often get adjusted by a semitone, because the two adjacent notes (the note from the key signature and the adjusted note) never actually happen side-by-side in real music (they're alternatives), so you can't (realistically) practice them both in the same scale. This means that people who want to play all the notes in order, and yet also want to practice all the notes they're likely to use, have to derive orders that include both variants.
In terms of learning both forms, you have to decide why you are learning a scale. Is it to learn about the relationships between the notes, or is it to develop finger dexterity? If a pianist is learning scales in order to understand the theory of music, and the relationships between notes, then I personally think he'd do better to abandon playing scales, and instead sing. If someone is learning for finger dexterity, then it's questionable whether the ability to play both the melodic and harmonic forms is any more valuable than the ability to play any other commonly-encountered pattern of notes. One might do better to turn to Hanon. But that's the uninformed opinion of an amateur.