maybe it was like that before you arrived, especially if someone has been tinkering with it. Fitting a new pump on the old cable would not cause it.
look for any motorised valves and count how many cores come out of them, and where they go, There may be a wiring centre where all the cables come together. If you can post some photos of the programmer, valves and pump, and the cables between them, it will help.
Youy can get wiring diagrams for boilers and their controls, some are not very simple, but we need to know the layout of the system you are working with, so don't pick one at random.
Or it might be that the boiler has an overrun timer on it to keep the pump running for a while after the boiler stops firing (this is to prevcent overheating). Mine runs on for 12 minutes.