I need to take another pic of the way the inside if set up. With cylindrical match cases I always do the rolled-up-striker-facing-outward and slide it into the tube. Then matches inside that. There's no way the match heads can contact the striking paper since facing the opposite direction. To use you can pull the whole thing out and wrap it around the match case for support. Sometimes I pull it up 1/2" and strike the top but honestly that's probably not the safest way (risks lighting the other matches).
“I'd rather have questions that cannot be answered than answers that can't be questioned.” —Richard Feynman