I dont think its so much a double standard as women wanting to have their own term that described the uniqueness of lesbian culture. Flesh pointed out the origins of the word lesbian, but I'm not sure if it was coined by lesbians thmsellves or applied to gay women by the straight population.

Homosexuality is the scientific definition to describe a same sex attraction, sex, ec. Being a gay man myself, gay also includes a culture and style unique to Ameriican Homosexual men and also implies behavior, habits and a certain sense of pride, which is now being adopted by other cultures as well. However, many men in India, for example will admit being attracted to men of the same sex but refused to be called "gay" because the term carries with it a connotation of being effeminite, and also the implication that you will seek same sex relationships. Men in India who admit they are only attracted to men will still marry women and have families, because they see it as a duty too.

Lesbians, recognizing the nuanced meaning of "gay", may wanted to have differentiate themselves from gay culture and so use the term lesbian. However, to show solidarity with gay men and the "gay" cause, they will also gladly be called gay.

I hope this answers your question! I recommend reading A History of sexuality by Michael Focault for more information on the history of how we interpret homosexuality.