The basic land types are Plains, Island, Swamp, Mountain, and Forest. Land types other than basic land types (such as Desert) don’t contribute to domain abilities.

Yavimaya Sojourner’s first ability can’t reduce the in its cost.

To determine the total cost of a spell, start with the mana cost or alternative cost you’re paying, add any cost increases, then apply any cost reductions (such as that of Yavimaya Sojourner). The mana value of the spell is determined by only its mana cost, no matter what the total cost to cast that spell was.

Domain abilities count the number of basic land types among lands you control, not how many lands you control or how many of any type.