Secondary question: Does The World Tree allows Reflecting Pool to produce mana of any color even if you don't have 6 lands, like a vivid land does even when it has no charge counter left?
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"De potentia juvenis somniabat, nunc de Mundo somniat..."
The reason a vivid land works that way is that it doesn't check whether the cost is currently payable. This also means that a tapped land (including a vivid land) still counts for this evaluation.
106.7. Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.
Example: Exotic Orchard has the ability “T: Add one mana of any color that a land an opponent controls could produce.” If your opponent controls no lands, activating Exotic Orchard’s mana ability will produce no mana. The same is true if you and your opponent each control no lands other than Exotic Orchards. However, if you control a Forest and an Exotic Orchard, and your opponent controls an Exotic Orchard, then each Exotic Orchard could produce G.
Interplanar Beacon works because the "Spend this mana only to cast planeswalker spells." restriction is ignored when determining what types of mana that could produce.
106.6. Some spells or abilities that produce mana restrict how that mana can be spent, have an additional effect that affects the spell or ability that mana is spent on, or create a delayed triggered ability (see rule 603.7a) that triggers when that mana is spent. This doesn’t affect the mana’s type.
Example: A player’s mana pool contains RG which can be spent only to cast creature spells. That player activates Doubling Cube’s ability, which reads “3, T: Double the amount of each type of unspent mana you have.” The player’s mana pool now has RRGG in it, RG of which can be spent on anything.
Secondary question: Does The World Tree allows Reflecting Pool to produce mana of any color even if you don't have 6 lands, like a vivid land does even when it has no charge counter left?
No, because it has no such ability at that time. Suppose you have The World Tree, a Reflecting Pool, and 4 Forests. They would then have these abilities:
The World Tree
The World Tree enters the battlefield tapped.
T: Add G.
As long as you control six or more lands, lands you control have "T: Add one mana of any color."
WWUUBBRRGG, T, Sacrifice The World Tree: Search your library for any number of God cards, put them onto the battlefield, then shuffle.
T: Add one mana of any color.
Reflecting Pool
T: Add one mana of any type that a land you control could produce.
T: Add one mana of any color.
Forest
T: Add G.
T: Add one mana of any color.
On The World Tree, the only mana abilities are #2 and #5. As it happens, #5 enables Reflecting Pool's ability #1 to produce any color of mana. But if you had one less Forest, none of the underlined abilities would exist at all; you would not have any lands with an ability to produce non-green mana.
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Why bother with mere rulings when so many answers can be found in the Rules?
For purposes of Reflecting Pool, a land "could produce" a given type of mana if any "ability of that permanent would produce [that type] if the ability were to resolve at that time", regardless of "whether any costs of the ability could or could not be paid" (C.R. 106.7). Thus, for purposes of Reflecting Pool:
Interplanar Beacon "could produce" any color of mana (because its last ability allows choosing any two colors) or colorless mana (because of its second ability).
The World Tree "could produce" only green mana if "you control" fewer than six lands, notably because in that case, The World Tree doesn't have the ability stated on its third ability. (Compare with Vivid Crag, which "could produce" mana of any color because its last ability "would produce [such mana] if the ability were to resolve at that time", regardless of its costs [C.R. 106.7].)
Secondary question: Does The World Tree allows Reflecting Pool to produce mana of any color even if you don't have 6 lands, like a vivid land does even when it has no charge counter left?
Yes, but not for the same reason as a vivid land.
The reason a vivid land works that way is that it doesn't check whether the cost is currently payable. This also means that a tapped land (including a vivid land) still counts for this evaluation.
Interplanar Beacon works because the "Spend this mana only to cast planeswalker spells." restriction is ignored when determining what types of mana that could produce.
No, because it has no such ability at that time. Suppose you have The World Tree, a Reflecting Pool, and 4 Forests. They would then have these abilities:
The World Tree