2002 Toyota Other Starting problem

What "ignites", and what exactly is the symptom? You haven't listed any test results or observations, but you did provide a clue. Since the engine runs when you push-start it, I assume the symptom is the starter doesn't crank the engine, but I'm usually wrong when I assume. How does it "start normally" if it's already running after a few miles?

If I'm right, do you hear a single rather loud clunk from the starter each time you turn the ignition switch to "crank"? You didn't list the car model either so I can't look up which starter is used.