Fallout 2 and Fallout Tactics have a few inconsistencies with Fallout lore here and there. The canon flag was originally meant to have 14 stars according to Tramell Isaac and JE Sawyer, but the FO1 version is only partially visible, and Chris Avellone mistakenly said that it's 13 in the Fallout Bible, and now Bethesda made the 13-star version officially canon.
So I'd say that now the 13-star version is canon, and the middle star stands for the Columbia Commonwealth, which includes Washington, DC.