Water still circulating through toilet even after I shut off the water supply!
My guess is that the shutoff valve (the first thing you showed in the video, near the floor) is not 100% shutting off the water. Hard water can cause this as deposits inside the valve can prevent the gate from fully shutting.

You could either replace the valve stem (can be done without shutting off hour house supply) or replace the shutoff valve entirely. I generally prefer replacing the entire shutoff valve with ball valves rather than gate valves, because I find they last longer and are easier to operate.

Here's an example of the kind I like to use: Ball shutoff valve example