DING DING DING ! We have a winner. State laws CANNOT define what a bow is. If it did .... then a crossbow IS a bow in Arkansas and the moment it is taken into Missouri it magically transforms into not a bow - and that don't happen. Its still a bow - just not legal in archery seaon unless you talka doctor into signing a permission slip.
You are right, it can not define what a bow actually is, just the "legal definition" of archery. Nobody argues that a crossbow has a type of bow affixed to a stock with a trigger mechanism affixed to hold the string .. even I'll admit there is a "bow" in there. But the crossbow isn't "just a bow" .. it is a bow that has been altered to be something else through the use of a stock and trigger mechanism. That makes it not archery equipment, or requiring archery skills to shoot.
So your crossbow is legal in Arkansas and not legal in Missouri during archery season, but even in Arkansas it is not archery to shoot a crossbow. They just let them use them there during archery season, not unlike allowing the disabled to use them in our state. It appears they even differentiate their kills as evidenced by Grizzly. I wonder of they break the kills down among by bows sauch as compounds and traditional? I'll guess not, that is probably because they are considered different weapons.