You may have a hi pressure pump mounted on the frame. That could be what you're hearing with the second "whir." I don't know specifics of the system, but I think the hi pressure pump remains inert until the truck is running and accelerating, then kicks in to compensate for the extra needed pressure for acceleration. Cycling the key on and off may be causing it to kick in.

As for what you may want to check, try the coil. If you have a multimeter you can test it. Here's the procedure outlined in my chilton manual-

1. Primary resistance is measured between the two primary [low voltage] coil terminals, with the coil connector disconnected and the ignition switch off. Primary resistance should be 0.3-1.0 ohms.

2. On dura spark ignitions, the secondary resistance is measured between the BATT and high voltage [secondary] terminals of the ignition coil with the ignition off, and the wiring from the coil disconnected. Secondary resistance must be 8000-11500 ohms.

3. If resistance tests are okay, but the coil is still suspected, test the coil on a coil tester by following the test equipment manufacturers instructions for a standard coil. If the reading differs from the original test, check for a defective wiring harness.

It goes on to describe other tests to be performed, but give the resistance a check and we can go from there. I suggest the coil because they can cause very funny issues when they start to go bad- random dying being a primary symptom.