Urborg's ability causes each land on the battlefield to have the land type Swamp. Any land that's a Swamp has the ability "{T}: Add {B}." Nothing else changes about those lands, including their names, other subtypes, other abilities, and whether they're legendary, basic, or snow.

If an effect such as that of Magus of the Moon causes Urborg to lose its abilities by setting it to a basic land type not in addition to its other types, it won't turn lands into Swamps, no matter in what order those effects started to apply.

Urborg, Tomb of Yawgmoth isn't a Swamp while it's not on the battlefield.

Land cards not on the battlefield aren't Swamps while Urborg is on the battlefield.