Two white films, two opposite chemistries
Limescale is calcium carbonate. It dissolves in acid and gives off carbon dioxide — that fizz is the reaction, and it is why vinegar, lemon juice and citric acid all work on a kettle. Soap scum is a different compound: when soap meets the calcium and magnesium in hard water, the fatty acid part binds to the calcium and forms calcium stearate, which is insoluble. It is the same reason soap will not lather in hard water — the soap is being used up making scum instead of foam.
And that is the trap. Calcium stearate does not dissolve in acid; acid tends to drive the equilibrium towards the insoluble calcium salt, so vinegar can fix the scum onto the glass rather than lift it. What dissolves it is the opposite: an alkali to saponify what is left of the fat, and a chelating agent to take the calcium away. In practice that means bicarbonate of soda, washing-up liquid or a soap-free alkaline cleaner, plus something like sodium citrate or EDTA for the stubborn layer.
So the same white film needs opposite treatments, and the shower is where the distinction is most often lost. Bathroom glass is not mostly limescale — it is mostly soap. Which is why a page that says "use vinegar on shower glass" makes things worse for most people, and why the glass goes white again within a week: the film was never removed, only rearranged.
Three whites, three answers
Chalky and dry, that a fingernail drags into powder, and that softens under vinegar in ten minutes, is scale. Citric acid at 100–200 g per litre of warm water, sprayed on, left ten to fifteen minutes, wiped with a soft sponge, rinsed and squeegeed. Do not leave acid on for half an hour and do not do it weekly: acid attacks the coating too.
Milky and slightly greasy, that a fingernail slides over rather than digs into, and that vinegar makes whiter, is scum. Bicarbonate paste, five minutes, circular work with a soft sponge — and for the stubborn band along the bottom, add a chelating agent and give it time rather than pressure. Rinse thoroughly, because alkaline residue is itself a film.
Hazy with a rainbow sheen, where the surface feels rough and a fingernail cannot raise any powder, is not a deposit at all: the glass has been etched, or the factory hydrophilic coating has gone. Etching happens with strong alkali over years and with abrasive powders; the coating goes with scouring pads and repeated acid. Nothing you can buy in a bottle restores either. Cerium oxide polishing is the professional answer, and on a large panel it usually costs more than the glass.