While I agree with you, technically Gen. 2:13 mentions Cush, which would be around the confluence of the Blue and White Nile rivers. Much interpretive gymnastics has been performed to place Cush somewhere in the Levant, but I find it unpersuasive. Also, “through” would be more properly translated “around,” as in 2 Chron. 21:9, where the Edomites surrounded Jehoram and his men.
13 The name of the second river is the Gihon; it winds through the entire land of Cush.