Find a diagram (try Ack's FAQ) and restore the wiring to original.
There's no way we can help you figure out what the previous owner rewired and/or bypassed - we can guess, but that's all it will be a guess.
My guess - if he did rewire - is that he wired the pump directly into the ignition feed so that it runs when the ignition is on, not very safe, but some cars are wired like that from the factory. Suzuki shuts the pump off if the engine dies for any reason - reduces the possibility of a leak and/or a fire.
If you can get the pump to run when you "mess with the wiring" then I'd say you have a bad connection somewhere - my suggestion is that you wire a test lamp to the pump connections and mount where it can easily be seen - verify that the lamp lights (and has close to 12V) the next time the pump doesn't run - you could be fiddling with the wiring when the pump is the problem.