TV screen in still photo: Why are there dark, blackish bands?

Yes, you're basically on the right track. The bands appear because the camera shutter is not synchronized to the vertical scanning of the CRT.

A fast shutter speed will only capture part of a scan. A shutter speed exactly equal to the vertical scan period will capture a full scan, but there still might be a narrow band (either dark or light) somewhere in the image if there's any mismatch in the timing.

To minimize the banding, use a shutter speed that spans several vertical scans, but this only works if the image on the CRT is static.

The only reason you didn't see "Fig 3" (bright stripe in the middle) is that your shutter speed is too slow. You have a much higher chance of capturing a dark gap in the middle, because those pictures all include a vertical blanking interval (the time when the beam is offscreen and "moving" back to the top) as well as all of the vertical part of the picture hidden by overscan. You can see from Fig 1b that your shutter time captures all but 1/3rd of the visible part of one field. To get "Fig 3" would require pretty precise timing, and your bright stripe would be nearly the full height of the display.