If it is a dual flight I am the PIC for the entire flight so that to me means that the engine doesn't get started until I am strapped in.
In any case the PIC is responsible to ensure the aircraft is airworthy. That IMO means at a Minimum I have briskly walked around the aircraft for a "nothing hanging/open/dripping" check and I have personally checked that the fuel level is sufficient and fuel caps and oil filler/oil dipstick are secure. This is pretty hard to do when the engine is already running.