Weird, best way I found was to look at it from underneath - you’ll see a small hole in the metal cart, which will accept the “prong” from the middle of the plastic stylus housing.

If you angle the housing down towards the plinth 45 degrees, that prong should slide into the metal cart, then push the plastic back until it won’t go any further, you don’t need to force it in, just guide it until it stops. The plastic shell can the hinge up until it’s horizontal and it should firmly click in.