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

The CRT Amusement Device uses a video display and has game-like elements, you could argue that makes it a "video game" (as opposed to a "computer game")