Chris, special thanks for providing the additional info.
I presume your extra start button was wired to the solenoid coil in place of the original wire. The original wire completes a rather involved circuit from the ignition switch to finally energise the solenoid, importantly as you now confirm along with a feed to the ECU. This sub circuit must activate a start mode, hence the reason for the sequence you describe.
The ideal would of course be to trace where the voltage drop causing the problem is occurring.
This has become a common failure yet to be resolved.
You have added still another clue in this rather baffling circuitry.
