And THE correct answer is ...............
This question is not like a simple mathematical equation where there is one correct solution (answer). Often in aviation there is no single "right" answer; who starts the engines falls into this category. It can be either pilot, or perhaps both required present for the start sequence. Convincing arguments can be made for every position.
It is admirable that you seek the "correct" and "professional" answer, but my position is there is not a single answer. If I were in charge; I know what I would do for the operation I was responsible for but that does not make my methods correct for every application worldwide every time. It depends on who the "boss" is and what there world view and values are; those foundations will drive the business decisions.
At my point it is not quite good to let passengers wait couple of minutes for captain to close the door, buckle himself into the seat and just then push start button. At least our passengers would send a complaint imediately for such behaviour - just because they are always in a hurry..
If your organization is in that much of a hurry your organization has lost sight of what is more important but your organization will not see it that way.
Years ago I was flying for a corporation and was on the ground in Teterboro New Jersey watching another "professional' crew get the jet ready. They must have been in a hurry because both engines were running and nobody was in the cockpit, the FBO's (Fixed Based Operators {Ground Handlers}) ground crew had the nose wheel chocked while the flight crew was busy scurrying around. Ridiculous, or as my teenager says reDonkeylous.