Sport vs War is also sort of feeling around an issue of what the combat part of the game focuses on, or is abstracted to.
War description implies the combat is informed by and has consequences for things outside the combat, but it largely a series of battles one after another. It implies a macro view of combat. Each battles is a little zoomed out in terms of impact.
Sport description implies "moments" in each combat are what matters, and combats have ~little impact on the outside campaign. This means a zoomed-in combat system that focuses on individual moments a lot.
Spectacle or Theater are somewhere in between. Combat as expression that informs the out of combat campaign, but also with some focus on individual moments per combat, as opposed to War where combat is almost a chore to be avoided.
Long-winded way to say that each type implies slightly different rules and levels of abstraction. You could run combat as any of these types, but the rules system you use will probably naturally lend itself to one of the styles.