The manchester mark 1 had a teleprinter as its output, and used a Williams tube as ram (https://en.wikipedia.org/wiki/Williams_tube). If the image on wikipedia is accurate, the checkers game only "displayed" itself incidentally, on the Williams tube, rather than actually outputting to the teleprinter. In your game, it would be like writing the current level to internal ram, rather than to the actual video memory. The Williams tube isn't really a TV-like device. It stores data on a CRT, but that CRT isn't visible to the user in general operation, as the read plate covers the "screen". Again, "first video game" is up to a lot of interpretation.
Also, saying that vector based video makes it not a video game is a little strange, given how common vector graphics were in arcades (eg Asteroids, Tempest, Missile Command) and the Vectrex
I'm not necessarily making the point that vector graphics based games aren't video games, just arguing against the parent comment against the claim that it wasn't a video game because it was stored in RAM.
I agree with the assertion that this was a video game because it was using a raster-based CRT for the display, even though the primary purpose of that display was for data storage not display.
Not necessary, you can just take an additional CRT and wire it in parallel to one of the Williams tube CRTs to see what's on the screen.
That's how the Manchester Baby did it (visible in the center of the image here): https://upload.wikimedia.org/wikipedia/commons/6/6f/Manchest...