The place is like Fort Knox, the security is tight, and they do have the capabilities to respond quickly to a bad situation.
This is a common misconception. Disney has no capability to respond to bad situations. None of their employees (even security) are armed. They rely on the local Sheriff's office. The safety of Disney world has little to do with their security and a lot to do with their price tag. Violent criminals just aren't going to pay the admission prices to go mug and rape people. The only people you really need to worry about in the park are deranged lunatics and tourists who snap. Deranged lunatics trying to commit mass murder are rare anywhere and tourists who snap typically don't commit murder. I figure if there is a deranged lunatic, I am just going to grab my kids and run and if a tourist snaps and tries to assault me, they will get a face full of OC.
Thugs who would kill you for the paper in your pocket are better off on the streets of Orlando. The only place on Disney property I would be concerned about is the parking lots (which we won't be using) and downtown disney (where I will be carrying).