Those who live in the area call Washington, DC: "DC". Those outside the area call it: "Washington". Many don't even readily recognize the name "DC". I moved to New York a year ago, people ask me where I moved from, I now say "Washington,DC" and that has it covered. When I first moved here I said "DC" and it confused some folks. ("Washington", alone, does sometimes make people think of the State; "Washington, DC" doesn't.)