Believoisie is a term of aspersion for theists used in some atheistic circles.
The earliest reference I can find to it is in a January 2005 Reason magazine article by Chris Lehmann, "Among the Non-Believers," reviewing the 2004 Sam Harris book The End of Faith: Religion, Terror, and the Future of Reason, a criticism of religious faith.
The faithful, meanwhile, take some understandable offense at this broad caricature of their mental capacity and ability to face life's harder truths. So each side retreats to its corner, more convinced than ever that the other is trafficking in pure, self-infatuated delusion for the basest of reasons: Believers accuse skeptics and unbelievers of thoughtless hedonism and nihilism; the secular set accuses the believoisie of superstition and antiscientific senselessness.
Lehmann is the source of your first quotation. The second is from Sam Kriss, a London-based writer and controversialist. Lehmann probably intended to use it as a nonce word, and there is probably no way to know who was the first person to coin it. It does not appear in the Corpus of Contemporary American English, nor even in the News on the Web Corpus, but as you note there are any number of blogs and other websites which employ it.
The word is a blend of believe, belief, or believer, referring to people who believe in religious tenets like the existence of God, and -oisie, extracted from the French word bourgeoisie. According to the OED, its earliest use in English is from around 1593:
Yet can nothing be done before ye Bourgeoisie being enformed of the fact agree also thereto.
Originally, it referred to the residents of a town (akin to burgess). By the time of the French Revolution it was associated with the urban middle class, and then to the tastes and conventions of the middle class, and then Karl Marx used the term to decry the capitalist class which control the means of production and exploit the proletariat.
The -oisie ending is not a true English suffix, but has been adopted in coinages referring to broad social groups. A dictionary example is booboisie, popularized by the critic H.L. Mencken (ca. 1920) as a humorous term for uncritical or small-minded people as if they were a discrete social class. There is latterly the hipoisie, for people seen to dictate tastes and fashion, found at least as far back as 1988, but used more recently perhaps because of the familiarity of the term hipster.