What decides the type of Hidden Power a Pokemon uses?

Yes, that is precisely how it works.

From the very website you mentioned, we see that, from generation II onwards, the formula to calculate Hidden Power's type is

where a, b, c, d, e, and f are respectively the mod 2 operation of the HP, Attack, Defense, Speed, Special Attack, and Special Defense IVs.

Depending on HPtype, Hidden Power is then assigned a type, like so:

• 0 -> Fighting
• 1 -> Flying
• 2 -> Poison
• 3 -> Ground
• 4 -> Rock
• 5 -> Bug
• 6 -> Ghost
• 7 -> Steel
• 8 -> Fire
• 9 -> Water
• 10 -> Grass
• 11 -> Electric
• 12 -> Psychic
• 13 -> Ice
• 14 -> Dragon
• 15 -> Dark

You might notice there is no Normal- or Fairy-typed Hidden Power.

This table gives you the highest possible IVs that give a specific type.
Due to the mod 2 operator, 31 can be replaced with any odd value, 30 with any even value.