Amethyst's ecclesiastical career is real and is the source of much of its modern reputation. It was the customary stone of a bishop's ring in the Latin church, and the association with sober spiritual authority followed from the name and from the colour, purple being the liturgical colour of penitence and of episcopal rank. This is a documented usage with a long continuity, not a folk claim.
The modern crystal-healing profile (calm, sobriety, protection against addiction, meditation, the third eye and crown), is a twentieth-century construction built on the ancient etymology. The step from not drunken, meaning a stone that lets you keep drinking, to a stone that helps you stop drinking is a substantial reinterpretation, and it inverts the original use. That inversion happened in the modern literature and is worth naming, because it is presented as continuity.
Amethyst is the February birthstone. It was on the 1912 National Association of Jewelers list and has survived every revision without change, which distinguishes it from months that have accumulated alternatives for commercial reasons. It also appears in the older breastplate-derived correlations, where the Hebrew ahlamah was rendered amethystos by the Septuagint, though the standard scholarly view is that ahlamah was more probably a red jasper, so even the apparent continuity of the name across two and a half thousand years may not be a continuity of the stone.
Modern metaphysical writing associates amethyst with the crown and third-eye centres. The chakra system as applied to gemstones is a twentieth-century synthesis and is not found in the Indian sources from which the chakra material itself derives.