Well, yes, the computer is truly turned off.
But, the power supply will have voltage available whenever the power cord is plugged in to the mains.
Not accurate to say it is turned on, and there is (virtually) no power drain of any kind, until the power button is pressed.
Power on boot does not happen until that occurs.
Pressing the power button connects a circuit which will sense if there is power available to the power supply. If the response is "yes", then the boot sequence will begin - and only then is power truly on.
If you want "no power available", then you have to unplug the power cord, or turn off the mains power, usually at a surge protector of some kind.