It might be the difference between Go-on (呉音, "sound from the Wu region") and Kan-on (漢音, "Han sound").
According to this, Go-on is:
introduced to Japan during the 5th and 6th centuries, when China was divided into separate Northern and Southern dynasties, go-on readings are possibly imported either directly from the Southern dynasty or the Korean Peninsula. This explanation is based mainly on historical reasoning: there was an influx of thinkers from China and Korea to Japan at that time, including both Buddhist and Confucian practitioners. However, there is no historical documentation to conclusively demonstrate that go-on readings are actually based on southern Chinese.
According to this, Kan-on is:
one of the sources of pronunciation of Japanese kanji. They were borrowed during the Tang dynasty (7th to 9th century), introduced by, among others, envoys from Japanese missions to Tang China. This period corresponds with the Japanese Nara period.
Kan-on is based on the central Chang'an pronunciation. The name Kan could refer to the Han dynasty, which also had Chang'an as its capital city.Furthermore, Kan has also become a description for all things Chinese, e.g., Kanji ('Chinese characters').
Kan'on partly displaced the earlier go'on, which were "just imitations of Korean imitations, but Kan-on were imitations of the real things."
Apart from the explanation of Kan-on above, here is an article that the Japanese government at that time often abolished Go-on and designated Kan-on as formal pronunciation of kanji. As a result of the policy of the government, we have a lot of Kan-on as on-yomi for kanji like [陰陽]{in-yoh}, but Go-on remains a little as on-yomi for kanji like [陰陽]{on-myoh} in certain field.
I quote the whole article in Japanese as follows.
陰陽師の読み方って、『おんみょうじ』以外に『いんようし』とも読めるのですか?
「いん」は漢音(七~八世紀ごろに遣唐使が伝えた中国語音)、「おん(おむ)」は呉音(六世紀ごろ、またはそれ以前におもに仏典を通じて伝わった中国語音)です。「陰」を「おん」と読む例は、仏教の「四苦八苦」の中に、「五陰盛苦(ごおんじょうく)」というのがあります。漢字音については、たびたび、「漢音を正音とし、呉音を廃する」旨のお触れがでました。そういう影響で、現代にいたるまで、漢字音の大半は漢音です。ただし、古くから伝わって、もうそれが漢語だと意識しないぐらい日本語になじんでしまった言葉は、あまり改まりませんでした。「おんみょうじ」もその一つだと思います。ことさらに漢音で「いんようし」と発音する人も時にはいたのかもしれません。それなら、「いんようし」でも正しいということになりますね。