Defend themselves, sure. But chase them down? I sure as hell wouldn't.

Some people would, some people wouldn't. It's not like you have to make a conclusive case for how likely it was, it's something that could have been true and it's not unreasonable to to have that though and believe it in the moment.

Personally I don't think even that detail should matter. If I was being chased by some dude who was certainly unarmed but seemed intent on violence, that should be enough, it doesn't take a weapon for your life to be in danger. If someone shot that person to save me, I'd hope they'd be in the clear (and then be charged for meth dealing, illegal handgun and whatever else of course).