Carnelian and its relatives sit at the centre of the breastplate identification problem, which is the oldest genuine gem-correspondence question in the Western tradition.
Exodus 28 describes the hoshen, the breastpiece of the high priest, set with twelve stones in four rows, each engraved with the name of a tribe. The Hebrew names are obscure and have been disputed for two thousand years. The first stone, odem, is a red stone and is rendered sardios in the Septuagint, carnelian or sard. That is one of the more secure identifications in the list; most are not.
Two ancient authorities give full identifications and they do not agree with each other. Josephus, in the first century, lists the stones in the Antiquities and also in the Jewish War, and his own two lists differ. Jerome, in the fourth century, gives another set in his letter to Fabiola on the priestly vestments and in the Vulgate. Later medieval lapidaries add more. The idea of correlating the twelve stones with the twelve months and the twelve signs appears in Josephus and is developed by Jerome, and that is the genuine ancient root of birthstone thinking, not the 1912 trade list, which shares almost none of the same stones.
The honest summary is that the breastplate scheme is genuinely ancient, genuinely a stone-and-tribe-and-month correspondence, and genuinely unidentifiable as to which minerals it meant.
Agrippa pairs sardonyx with the fixed star Antares in the Behenian list, which is a separate Renaissance scheme with its own logic, and pairs ruby or garnet with Aldebaran.