An interesting PDF given above. Someone put a lot of effort into producing it and woudn’t it be great to have one for every lathe.
If you pull the cross slide handle towards you with one hand and hold it to take up any slack between it and the saddle, then use the other hand to move the cross slide itself. The movement you get will be the nut both its wear and slackness in the mounting screws. I think you’ve got a screw loose

The as a different operation if you pull the handle while pushing the slide you will take up the slack in the nut. Keeping this tension on move both hands together back and forth and this will show the slack in the cross slide screw mounting registration.
I think in picture 156 of the pdf you see a ridge on the screw that forms this registration by being trapped between the almost circular boss and the saddle. So I was wrong abut the screw and handle controlling this. Idealy you need some brass shim washers here to take up all but a wisker of slackness.
I see one article about replacing the nice ball handle because the huge cap head screw gets in the way. Just replace the cap head with a normal hex head, with the head skimmed down a bit to minimum thickness, and if available replace the washer with a thinner crincle one too.