Have you tried something more acidic or did you go straight for vinegar? The thing is, that white residue really is limescale, like they said, but the trick is not letting the solution dry too quickly on the glass.
Put pure white vinegar in a spray bottle and apply it generously to the stain. Let it sit for about 10-15 minutes without touching it, then wipe with a microfiber cloth or paper towel in circular motions, not in straight lines. If the stain is stubborn, repeat the process. The important thing is keeping the area wet while you're cleaning, because the vinegar needs time to dissolve the limescale.
If you don't have vinegar at home, citric acid works too (like lemon juice or dissolved in water), but vinegar is more practical because it comes ready to use. And don't worry - glass is durable, you won't damage it with a cloth or spray, as long as you're not using overly aggressive products like strong acid or steel wool.