Got to say, I've never thought about that before. I suppose it *is the street name, although "East" and "West" do not appear on the street signs. You do use them, however, when addressing envelopes and packages for the mail.
The dividing line between East and West for Manhattan addresses is Fifth Avenue. West of Fifth Avenue, the addresses go up from the Avenue to the Hudson River (the western shore of the island) and east of Fifth Avenue, the addresses go up toward the East River (the eastern shore).
In conversation, some streets are referred to as West or East. For example, West Fourth Street is "West Fourth Street." On the other hand, as far as I know, West 20th Street is just "20th Street." You would add "West" with a little emphasis, or in a second statement, if you wanted to make clear which side of town you were referring to.
For your purposes, I would, as you correctly infer, treat "West" or "East" as part of the street name.
