You are referring to a hybrid car (such as a Prius), correct?
That being the case, in a general sense, the"Start" button (on the dash as a rule) initiates the engine start sequence from the "Off" condition that was set when you removed the key (or a key fob) when exiting the car.
The "start" signal for re-starting the engine, from a stop in motion such as at a stop light, is initiated when you press on the accelerator, i.e., a sensor in the accelerator electrical/mechanical arrangement is triggered.
Other factors (e.g., battery condition, load on the power system such as A/C usage, etc.) may play a part in the sequence, as well.