We've used Blue Moon to mean the second Full Moon in a single calendar month.
In March 1999, Sky & Telescope magazine published an article about Blue Moons by Philip Hiscock who has studied the folklore and history of the expression. In that article, Hiscock traced the many meanings of the expression over the centuries, but noted that the "two Full moons in a single month" meaning couldn't be explained satisfactorily.
In the May 1999 issue of Sky & Telescope, there appeared a follow-up article which proved that Sky & Telescope had in fact created the current meaning by mistake in an article published in March 1946. The author of the 1946 article had misinterpreted a page of the 1937 Maine Farmers' Almanac.
By studying copies of the Maine Farmers' Almanac dating as far back as 1819, the authors of the May 1999 article showed that the compilers of the Almanac used the term to label the third Full Moon in a season which has four.
We have calculated the dates of this type of Blue Moon for the 20th and 21st centuries and put them in a list for you to browse.
It's a delightful irony that Sky & Telescope, in publishing an article in March 1999 on the history and folklore of Blue Moons, should turn out to be celebrating a "tradition" which it inadvertently created in an article 53 years before!
So which definition is "correct"? The authors of the May 1999 article admit,
With two decades of popular usage behind it, the second-full-Moon-in-a-month (mis)interpretation is like a genie that can't be forced back into its bottle.
And Charles A. Federer, Jr, the founder of Sky & Telescope magazine, adds,
Even if the calendrical meaning is new, I don't see any harm in it. It's something fun to talk about, and it helps attract people to astronomy.