It is important to note that when a jewel is manifesting one form, it is not able to do others, at least for now. He is also only capable of using one jewel at a time at the moment and will come equipped with one type of jewel once in each fight, even in longer fights. Even when he begins to fight, he is passively creating a new jewel, but it'll take a while for it to finish in order to be used right away. It typically takes about twenty or so in-game minutes for a fresh new jewel to be created.
Here is a brief explanation of each manifestation:
Spell- User is capable of taking a jewel's magical ability. This can be something as simple as a fireball to increasing Alaric speed. At rank 6, Huxley can use a spell up to twelve times before the jewel will break.
Artifact- User gains a tool which the jewel manifests. Can be a weapon, defensive item, or something seemingly useless. It drains the jewel very slowly; if a gem is used throughout a whole fight, it will break at the end. Can or will not replace Huxley's jewel rapier depending on the item.
Giant: Jewel giants deplete everything within a jewel. It instantly destroys it, but is capable of summoning a very large giant sleeping within the the gem. The word "giant" does not necessarily mean that what summoned is aggressive or offensive. In fact it refers only to the size, whether by length, width or height and every jewel giant is different in nature.
Spell vs Rank Growth