Power Source Mode does not Change: Description
WARNING: This page is about a different car, the 2011 Toyota Corolla. However, it is still accessible from the selected car via links, so may be relevant.
When the engine switch is pushed with the electrical key in the cabin, the main body ECU (instrument panel junction block) receives signals to switch the power source mode.
HINT:
To allow use of the Techstream to inspect the push-button start function when the engine switch is off, repeat opening and closing any of the doors. Opening and closing a door establishes communication between the Techstream and the main body ECU (instrument panel junction block). (Opening and closing a door can also be simulated by operating a door courtesy light switch.)