While I agree with you on the result, the rules snippet doesn't fit for the question. "A card with a basic land type", the artifact lands have no basic land type.
Yeah, I know.
I'm already searching for a fitting rule. But I don't find the part that states that lands do get the color identity due to the mana they can produce (exluding "any color").
Edit:
nonsense, I'm stupid.
903.4. The Commander variant uses color identity to determine what cards can be in a deck with a certain commander.
The color identity of a card is the color or colors of any mana symbols in that card’s mana cost or
rules text, plus any colors defined by its characteristic-defining abilities (see rule 604.3) or color indicator (see rule 204).
Example:
Bosh, Iron Golem is a legendary artifact creature with mana cost {8} and the ability “{3}{R}, Sacrifice an artifact:
Bosh, Iron Golem deals damage equal to the sacrificed artifact’s converted mana cost to any target.” Bosh’s color identity is red.
____________
And the explanation why only color-fitting basic lands can be included:
305.6. The basic land types are
Plains,
Island,
Swamp,
Mountain, and
Forest. If an object uses the words “basic land type,” it’s referring to one of these subtypes.
An object with the land card type and a basic land type has the intrinsic ability “{T}: Add [mana symbol],” even if the text box doesn’t actually contain that text or the object has no text box. For
Plains, [mana symbol] is {W}; for Islands, {U}; for Swamps, {B}; for Mountains, {R}; and for Forests, {G}. See rule 107.4a. See also rule 605, “Mana Abilities.”