I've been meaning to write an answer to your question since you posted it, but I didn't get around to actually writing it. Well, I hope that you're still waiting for an answer.

I have a few problems with the history-sections of that Wikipedia page. The most basic would be that it doesn't paint a good picture of the miasma theory at all, completely reducing it to "miasmatic vapors are the only thing that can make people sick", disregarding its connections to and interactions with humorism. Beyond that, it shoehorns classical and medieval ideas of contagion into the development of the germ theory, even though that is technically not correct. Finally, the article largely disregards that, up until the 19th century, miasma theory was actually superior – logically and empirically – to germ theory.

1. Miasmatic theory

Even though "bad air" was the central disease-causing factor within miasma theory, it was in no way the only one. Miasmatic air alone was only thought to be the base cause of epidemic diseases. There are still more factors that played into outbreaks: most importantly, a population needed a certain "epidemic constitution" for an outbreak to occur. This "epidemic constitution" simply referred to a humoral predisposition for certain diseases.

While the above may seem nitpicky, there is one thing missing in the article's description of miasma theory that is just wrong.

Such infection was not passed between individuals but would affect individuals within the locale that gave rise to such vapors.

Contagion may have played a minor role in the context of miasma theory and humorism, but it was not disregarded. Guy de Chauliac, physician to pope Clement VI, theorized in 1348 that the Black Death could be contagious in certain cases and advised Clement to avoid sick people. All of this while still firmly believing that miasmatic vapors were the prime cause of the epidemic. And he was not the only one who tried to avoid contagion: basically everywhere the Black Death arrived, local authorities set up quarantine measures to avoid or at least minimize contagion. By the 13th century, quarantines had been used for centuries: Gall of Clermont advised Desiderius of Cahors to close certain roads to areas

Miasma theory and contagion have historically never been seen as mutually exclusive in the context of Western European medicine.

2. Early "predecessors" of germ theory