No, you're just starting from the wrong point of reference. The center of the world is clearly somewhere around Russia or Egypt. That makes Europe the west, Asia the east, and the Americas non-existent.
According to the degrees of longtitude, the Americas are part of the Western hemisphere.. add in britain and the tip of spain. The rest of the world falls in the Eastern hemisphere. Sure, from different points of views different places will be west or east. But it's easier for the sailors and plane pilots to set their prime meridian 0, at Greenwich.. and declare one half east.. the other half west.
Yep, there's a day change, but that's the only way you can make sure that time across the globe generally resembles where the sun is. Because we use a timezone system, think of the earth not as a globe but as an unrolled roll of paper.