Dorothy, n. More fully friend of Dorothy, this is a slang term for a gay man, after Judy Garland’s character in the 1939 film The Wizard of Oz.
I think this is misleading. Dorothy is not a short form of friend of Dorothy, they are two different expressions used differently. “Dorothy” is used as a form of nickname address or reference to a particular gay man, as in the OED’s “When’s Dorothy and Toto getting here with the chest of drawers?” The phrase friend(s) of Dorothy refers to gay men as a group, or a single gay man as a representative of that group ("He’s a friend of Dorothy [if you know what I mean]"), without regard to the individuality of the person under discussion. This should become clear if you try to imagine talking about gay men as “Dorothys.”