I think nowadays, “warbow” is a generic term for the English bows like the Mary Rose Tudor-period weapons. But, I think the term war bow applies to any bow used in warfare, and there are many styles and cultures that developed and used them, as others have pointed out here. Any bow used to shoot even one arrow in a skirmish or battle should then be considered a war bow, even if it was a hunting weapon someone shot some other guy with.
I don’t know how far back the term goes, or if it was ever used by bowyers historically, but feel it is really a marketing “branding” term, like the term “horse bow,” and that kind of pernicious thinking really harms everyone. There were mounted Medieval English archers using standard yew longbows from the saddle, but you would never call those bows horse bows.
Cross bows seem to have avoided all that. There were the big, powerful kind used to kill the enemy, and elegant sporting crossbows, even stone bows for killing small game and birds, but I have never heard of a “war crossbow.” A cross bow is simply a cross bow.
I think “military bow” is a more appropriate term for bows crafted specifically for a given war or campaign, fortification protection, guard duty, sieges, etc., and differentiates the intended purpose of a military weapon from a sporting or hunting weapon.
Dane