It is all SiO₂
Quartz is silicon dioxide, trigonal, hardness 7, with a molar mass of 62.1 grams per mole and a composition of 51.2% silicon and 48.8% oxygen by weight.
Every variety name below refers to that same species. None of them is a separate mineral, and none of them appears in the IMA list as such.
Colour varieties
Amethyst is purple from iron impurities and natural irradiation. Citrine is yellow to orange, and most material sold as citrine is heat-treated amethyst. Rose quartz owes its colour to microscopic mineral inclusions rather than a dissolved impurity.
Smoky quartz is darkened by natural radiation acting on aluminium impurities, and can be produced artificially in a way that is difficult to detect.
Cryptocrystalline varieties
Agate, chalcedony, jasper, onyx and flint are quartz with crystals too small to see. The distinctions between them are textural and, in the trade, frequently arbitrary.
Banding makes agate; opacity and iron staining make jasper; the line between them is drawn differently by different dealers.
What this means when buying
Because the variety names carry no compositional meaning, they carry no guarantee either. Citrine and heat-treated amethyst are the same material by every test available to a collector, and the price difference between them is a matter of disclosure rather than chemistry.
Ask what the material was before it reached its current colour. For quartz specifically, the honest answer is often heat or irradiation, and a dealer willing to say so is worth returning to.