I, too, call BS on the story. The article I read indicated he was using a laser sight on his .380 auto, which he claimed help him get a clean kill. This makes me think both he and the coyote were standing still. (Confession: I've never used a laser sighted firearm.) He may have killed the coyote, or not but I think he's exaggerating the story a bit to puff up his tough Texan cred. And no, I've never run with a gun but I do own guns and grew up in a hunting family and am not inherently anti-gun.
The .380 hollow point is an odd round for a snake, if that's what you're into. A more suitable round for snakes would be a .22 rat shot. I’m thinking he carries a compact .380 for protection from two legged varmints. The laser sight is more for dork effect, kind of like having a big airfoil on the back of your Honda Accord.
Your biggest risk from a rattlesnake is from stepping on it. If you know it’s there, then you can chase it off the trail or go around it. If you step on it, it’s a bit late for the gun.
Deployed servicemen in combat zones (like Iraq) are required to run with a gun. (Or at least all of my deployed running buddies were.)