Duct insulation vs pipe insulation — what's the difference, and which one causes wet patches on my ceiling?

Good question — they get muddled constantly, and the fix is different for each.

Pipe (line) insulation is the foam lagging around the copper refrigerant pipes running between the indoor and outdoor units. Its job is to stop those cold pipes losing energy and to stop them sweating where they pass through warm/humid space — see pipe-lagging. When it perishes you get dripping along the pipe run, often at the outdoor unit or in a riser.

Duct insulation wraps or lines the ductwork that carries the cooled air. It does two things: keeps the air cold on its way to the room (efficiency), and — critically for your problem — keeps the cold outer surface of the duct above the dew point so it doesn't sweat in the ceiling void. A gap, crushed section, or stripped-back duct wrap lets the metal go cold, humid void air condenses on it, and the water drips onto the plasterboard → exactly the brown ring you're seeing.

So your stain is almost certainly duct-side, not the refrigerant pipes — either a break in the duct insulation right above that vent, or the runner-up cause: a blocked condensate drain letting the tray overflow. Both live above that ceiling. It's not a refrigerant/gas issue and shouldn't be sold as one. Get the void opened at that vent and the insulation and drain checked — a standard part of AC repair and maintenance. Good insulation is rated by its R-value; manufacturers such as Rockwool publish the figures for duct and pipe products.