i came to suggest the fuel pump, and then after I started reading your comment I was literally going to suggest leaving it turned on so the pump could prime before trying to turn over the engine, haha, so you're on the right track.
I wouldn't try listening for the pump, I'd be surprised if you could hear the fuel pump anyway, they're super quiet. it's a slow siphoning of a small amount of fuel, and I'm not sure what the diagnostics sound for that would be.
is there any difference in the ignition sound if you keep the key turned on but not igniting for 10-20 seconds before trying to turn it over?
leaving it in the on position before turning it all the way over should give the fuel pump time to get fuel to the engine if it's failing and isn't delivering the fuel at the rate it should.
also, I always have one of those $30 code checkers from Walmart in my car.
they can point you in the right direction, and I've saved a lot of time using one of those.