In the post below I said it was ruled that you must declare Cavern of Souls' colored mana to make the creature uncounterable.
After a week or so and review among different judges and players, the ruling is reversed.
The NEW ruling is this: The only time it matters is if you're casting a creature of the named creature type (otherwise you can only use the colorless ability). When you do cast a creature of the named creature type, the assumption will be that you're choosing the (generally) most favorable ability, which in this case is to use the colored mana therefore making your creature uncounterable.
How will we handle this? Basically...the same. The only time colorless goes into a creature of the named creature type is if the player casting the spell says so. Otherwise, it's assumed he chose the (generally) more favorable "make colored mana and it's uncounterable."
Clear as mud? :P lol.