very dodgy piping

I think that it's the frost dump valve for the boiler. You'll have to close the valve and it should all be back to normal. Google Truma Frost Dump Valve.

Frrom what we can see, they look like two different diameters. So the open one is narrower and it flowed water when I ran water from the sinks. It ran out while I was running the tap, rather than afterwards, as far as I can tell. The boiler was not on at all. The van is a 98 Hymer Camp Swing. Guessing the boiler is the same age.
I'm totally confused

sounds like the dump valve in the boiler. when the tempreture in your habitaion area falls below a certain degree, the dump valve opens and gets rid of the water in your boiler to prevent it freezing. should be a spindle like valve at the front of your boiler, at least on mine it is, that you pull up and it closes the valve.

Look out for a clothes peg near the boiler somewhere too just in case. That is usually used to stop the boiler dumping water rather than a cable tie. I discovered that one day in my own 1998 MH when I dislodged it by accident in the garage and wondered where all this water was coming from.

To refill the boiler make sure you have water on board and just run your hot tap until air stops coming out of the tap. Boiler is it a closed system. Like your own hot tank at home when you use hot water it is topped up again with fresh cold water. Open all of your taps and run them to get all of the air out. You will probably hear your boiler refilling.

The thinner pipe could be from the sluice valve for draining the water system. Very often, people leave this open over the winter months.
The water pump is only activated when you turn on the tap.
After you close the tap, the pump stops after it has pressurized the system.
The pressure relief valve is a safety feature, which only releases water should your pump malfunctions and over pressurized. This stops your pipes from bursting.
Also as the hot water expands in the boiler, the safety valve may drip a little. Again this is relieving the excess pressure in the system.

The yellow lever in the photo is my sluice valve in the closed position. Flip it over and it is open and if I turn on a tap when open, the water is dumped.
I hope that this helps.
Malcolm

Thank you all so much. My head is exploding with all the new information over the past few weeks!
We tried to replicate the problem last night but only a few drops came out, nothing like the previous night. Going to have a nose around those pipes in the boiler closet. I didn't know which bit was the lever, so thanks for the pic - I know what to look for now.

The pipe with the cable tie on it - do I need to take it off to allow it to output water when necessary to avoid damaging the boiler?

Thanks again

You should find what the pipes are connected to before you disconnect anything.
See if you can post s few photos of where they ate connected to.