You're going to end up with more Figureheads.
The thing is, if your President is off doing war-stuff, then that means he isn't at home doing his normal Presidenting stuff. But he serves an important function, and has a lot of power, and that creates a power vacuum that someone else will fill while your President is away at war. So you're going to end up with Power Grabs, where the President's normal authority is slowly eroded each time he has to go off and fight people instead of doing his normal job. Eventually you just have a figurehead position where he doesn't do much most of the time, and only has significant authority during wartime.
This applies to other World Leader-types as well, even though they will in practice have different responsibilities, you'll see the same sort of power restructure over time.
So to answer your specific questions:
How would it change the course of history?
On the one hand this would create more of a power re-distribution, rather than a fundamental change. So I don't think you'd see many direct changes caused by this. On the other hand, consider the butterfly effect - small changes in human political power structures could cause extremely large changes over time. So the world could be very different even with only minor historical changes at each step.
Would presidents be more or less likely to engage in war?
Even in our political systems, presidents don't have as much influence on war as you might think in most modern first world countries. They probably have the most individual influence, but that's dwarfed by institutional, party, or societal pressure when determining if a country goes to war or not.
Would the people be more likely to elect military figures knowing that if war broke out during in next 4-8 years that the president would be more directly, tactically involved?
Probably, but since your presidents will end up with less non-wartime powers, this won't be as big of a change as you might think.