How to find out what style of shifter is in 3 speed with overdrive on the floor

You might need to have another look at what's loose.
In my experience, the looseness is usually at the interface between the shift lever on the side of the cover, and the studs coming out of the cover. Over the years the nuts have come loose so often, and then been retightened that the flats on both of those are no longer tight. I have tried several cures but the best is to take the cover off, pull out the studded levers, coat them with silicon O-ring lube, then slip them back in. Next, reinstall the levers, correctly oriented, then put a couple of drops of Red locktite at the interface, to fill up the cavity. Finally a couple of drops of Blue Loctite on new factory serrated washer-nuts, torque it up and set it down with the inside face of the cover horizontal and facing away from the bench. Come back every 10 minutes and shift the levers to make sure the red loctite has not migrated past the O-ring lube. After three times. it is ready to re-install.
Do not be tempted to use regular hex nuts with washers or split loc-washers,sometimes called "spring-washers", because they will come loose. In a pinch, you can reuse the factory serrated nuts.
And that solves that problem for many years.
All the other slop that may exist, on the street, IMO, is not a big deal with that trans......... because it will spend oh say 95% of it's life shuttling between first and second,lol.
unless the shift-handle is flopping around,lol, that's just so annoying.