99 Honda Accord won't start cold

What happens when you try to start the car and it does not start: when you turn the key to Start the engine does not turn over [starter does not engage], or you turn the key to Start, the starter turns over the engine [spins the crankshaft, etc.], but the engine does not “fire up” and run? This distinction, plus the sounds you hear or do not hear, the dashboard lights you see or do not see, are important facts.

My 1999 Civic, when the fuel level in the tank is low, sometimes cranks for a long time but does not “fire up” and run. If I turn the key from Off to Run several times, making the fuel pump run each time, and then turn the key to Start, it fires up and runs fine.