Very likely its the neutral sensing switch on the tranny BUT I had a problem similar to yours, I didn't mind using the short piece of wire to hot start the car (it was VERY intermittent!)
But my wife uses the car occasionally and she is not capable of doing that so I replaced the neut switch. Then, Lo and Behold! the problem still was there
It has since been determined that my starter switch (so-called ignition sw) is probably the culprit, so don't rule that out too as potential source of this kind of problem.
It isn't your starter, after all the car will start OK when you use the jumper wire, doesn't it? so its not the starter motor!