Is there such a thing as a first ANYTHING war? Warfare is constantly evolving, and the contrast between the Crimean War and the ACW/French-Prussian war was just as sharp as that between the ACW and WW I, which was in turn just as sharp as the passage from WW I to WW II, and so forth.
Each war will see the introduction of new tactics and innovations, and each war will see some of the tactics and innovations of the last one become obsolete. That's the way wars are.
For example, look at a truly modern war, important things in it : the radar, jet planes, missiles, etc - all elements that were introduced to warfare in WW II. Does that make WW II the first modern war?
But wait! WW II wasn't really a break from anything - Carriers, planes, tanks - all were first developed for military use over the course of WW I, so shouldn't THAT be the first one?
But then trench warfare warfare was developed in the ACW, so WW I wasn't really a break from previous wars, more of a continuation...
(the list goes on and on, I suspect all the way until the first man hit his neighbor with a club to get a better share of the meat).