yep,   the push button merely  tells the computer to  'do it's thing'...  and   sends the crank signal to the solenoid.  when  RPM  hits a  certain number, (or maybe  amp draw drops)  the  'puter  disengages the starter.

My 2005 Ford triton 5.4  drives me nuts with it in  cold weather starts.. when it 'just about  lights up'...  it quits  cranking..    
If it  takes  more than 3 attempts,  it's  flooded!!