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

Yavimaya, Cradle of Growth isn't a Forest while it's not on the battlefield.

Land cards not on the battlefield aren't Forests while Yavimaya is on the battlefield.