It is normal to have a VERY SLIGHT amount of movement, but when I set mine, I always start at zero play. Any slop after that is due to the wear in the bearing. The spindle on a Woods is set up exactly like a wheel bearing on the front of a traditional car. Instead of a castellated hex nut to adjust preload, Woods made use of the fact that the split/ tapered bushing type of belt pulley bore can be made to lock tightly anywhere on the spindle shaft. I use a rather hillbilly method of adjusting mine, but in case it is helpful- here ya go, use or not at your own risk;
First, at the top of the spindle above the deck I saturate the little bolts, the inner part of the spindle shaft, and the split bushing with my favorite brand of penetrating magic, which is whatever I have handy.
Second I raise the deck of the mower with the tractor 3pt as high as it will go, and either loosen or pull the belt.
Third, I place a block of wood, or a piece of stovewood, etc. under the spindle I will be working upon. It doesn't matter whether the blade is attached or not.
Fourth, I let the deck down until the bottom of the spindle shaft bears upon the wood. No reason to get carried away here, you just want SOME weight resting through the shaft on the wood. IF THE SHAFT IS UNSUPPORTED, IT COULD FALL OUT OF THE SPINDLE LATER IN THE PROCEDURE.
Fifth, I back out the small screws that are holding the bushing to the pulley.
Sixth, I put the same screws (BEWARE, THESE LITTLE SCREWS AGE POORLY AND ARE JUST BARELY ADEQUATE WHEN NEW. IF THEY ARE MARGINAL, GET SOME SPARES AT THE STORE)that I pulled from the bushing into the second set of bushing holes. These holes are threaded in the bushing INSTEAD of the Pulley, and when you tighten them against the pulley face it pushes the pulley downwards and separates the pulley from the tapered bushing.
Seventh, I alternate between the two screws tightening until it pushes the pulley downwards free from the bushing. Care must be used to ensure that you go back and forth between the two screws.
Eighth, the pulley should be down, and the bushing should be relatively free to move up and down the shaft. I take my Grandmother's small "Rocket" claw hammer and VERY lightly tap the bushing downwards to take all the play out. It should move by approximately the same amount as the play that was noticed during our initial survey. Lightly is the key, just a little harder than you would hit a pecan to crack it, maybe about like you would drive a 4 penny finish nail in soft pine. The square key may need to be driven down at this point, and those keys are a weak spot. Keep a key or three spare in the box, they will be needed SOME DAY.
Ninth, I put the screws back in the original holes (The ones that are smooth bore on the bushing and threaded in the pulley) and pull the pulley back up upon the bushing. Easy does, little by little, alternating between the two bolts. When the bolts are tight and pulley is back up, we are safe to raise the deck.
Last, I raise the deck back up, grease the daylights out of it, and re-install the belt.
That middle one that you are fighting is a tough one, make sure the zerk is lined up where you can get your grease gun on it.
I generally grease my deck about every other tank of fuel, which for me works out to one long day of mowing.
Good luck, I have mowed around 20 acres with mine and had a belt last for three years, however I will go through as many as six of those friction discs in a year, mostly through operator negligence or episodic distraction.
Keep well, Jeff