If I get this right, your current pump is actually a double pump - two entirely separate systems, is that correct?
Usually the heating circuit is on a 'call' wire from the thermostat. I'd check the wiring diagram for your pump, find out which wire/terminal that is to your pump and disconnect the wire. If the pump stops, you know there is a wiring fault to the pump (rather than a fault in the pump). Ditto for the other circuit - that will be on a thermostat on the tank I would have thought.
The only system at all similar to yours I had, the heat exchanger pump was in the boiler itself and the thermostat was on the tank. When the tank cooled, it signalled the boiler to fire up, that also started the heat exchanger pump (which was part of the boiler).