My view of Cat ratings is simple - What the testing labs say it means. Any attempt to simplify the standard basically changes what the standard says.
However, this is how I think about it.
First, any equipment that has any connection to mains voltage could deliver a possibly lethal current due to an unanticipated failure. This includes a circuit powered by a brick, wall wart or whatever.
Second, the less things between what you are doing and the step down transformer out on the street, the more possibility of lethality, the higher the Cat rating is appropriate.
Third, user mistakes can and do occur. Higher Cat ratings recognize the possibility of user error and the greater need to protect the user. Example old style not guarded pin connectors or banana plug connectors could pull out of the socket and cause injury.
Fourth, Higher CAT ratings are protective but don't guarantee safety for an inattentive or careless user.
Fifth, the CAT rating of the system is the lowest CAT rating in use. A CAT III meter with Cat II probes is a Cat II system. Carelessness or lack of skill while probing high voltages can lead to a dead CAT no matter the CAT Number. Basic Care is to assume lethal voltage is everywhere until proven differently.
You can die of mains electricity and not blow a fuse. Therefore concerns about current ratings of fuses in prior posts are misplaced. Lethality has to do with whether the current, the length of time the current is there and how much can pass through a major organ. Voltage high enough to cause dielectric breakdown of skin, reducing resistance, increase critical currents. Fuses are there to prevent fire. That's a different topic.
Nothing that I just wrote is absolute. Read some books. Simplifying creates a real possibility of leaving something out. Here is a nice wiki page.
https://en.wikipedia.org/wiki/Electrical_injury