Well, I'm running out of ideas on this one. It's certainly very odd. I still think it's an external control fault, because of the fact that the cylinder thermostat which (if wired correctly) shouldn't have a live supply from anywhere, is switching the pump / boiler when you adjust it.
The absurdity of this could be illustrated like this. If you turned off the main electrical supply to your house at the consumer unit, then when into the kitchen and switched on the kitchen light would it come on? No, it wouldn't happen (unless it was wired incorrectly). What you have is the same situation. The electrical supply from the Heatlink to the cylinder thermostat (terminal 6) is off, yet, you turn on the thermostat and the pump runs. Shouldn't happen.
So, to me I still believe the wiring is wrong somewhere. The crux of the matter is, where is the cylinder thermostat getting a live supply from when it shouldn't be. So I would start with checking the cylinder thermostat is wired as it should be, according to the link to the diagrams I sent earlier. Not just that the correct three switching wires are going there, but that they are corrected to the right terminals.
Then there is the other matter of:
So basically the pump is running without calling for heat?
Is the pump wired to the terminals provided for it at the boiler? (this is different to the diagrams because your boiler requires the pump to run for a while at the end of the heating sequence to cool the boiler, and so is wired directly to special terminals for it at the boiler)