In discussion of orbit boxes, I don't think I've ever heard anyone talk about the size of the box in the nadir (your Z) direction. The most important dimension of the box is the east-west (is this your X?) direction, because that's where your neighbors are, and intruding into their boxes is what you're supposed to maneuver to avoid. Controlling the Z direction is important mainly because changing the nadir distance is precisely how you make east-west maneuvers.
That is, if you're not exactly at geosynchronous radius, then you're necessarily drifting east or west, depending on whether you are slightly closer (and thus faster) or slightly farther away (and thus slower). The larger the absolute difference between your actual nadir distance and the idealized synchronous one, the faster your subpoint longitude drifts. You get to pick how fast you want to move from one side of your orbit box to the other, and set your temporary altitude accordingly, until you return to your nominal orbit and start drifting again (slower, in the other direction) under the natural perturbations.
The usual cadence of operation is a small burn a few times a month, which is sufficient to stay inside the orbit boxes as currently assigned. You could choose to do several burns every day, but that would cause a lot more wear and tear on both vehicle and crew, and it's overkill for this problem. The 20 to 30 km distances you found sound reasonable to me. For a specific example of a larger difference (100 km, in order to drift from one geostationary slot to another), see this NOAA/NESDIS page about moving GOES-16 to replace GOES-13.
The specific perturbation of greatest consequence here is the fact that mountains are denser than water or air, so mountains exert more gravitational attraction than the same volume of water or air. Therefore, satellites placed in supposedly geostationary orbits all drift slowly towards one of two specific longitudes, one roughly centered on the Himalayas, and one roughly centered on the Rockies & Andes. Usually, operators slowly drift down this slope potential in the oceans-toward-mountains direction from one side of their box to the other, then change radius (Z) to drift back the other way. It is in principle possible to make your Z exactly the right value to cancel this perturbation, but there are many other influences (moon, sun, planets, sunlight, solar wind, additional terms in the earth gravity field, etc.) that also cause east-west drifts, and they vary differently with time.
You could try to cancel all of them, but there's a technology limitation: in general, thrusters have a minimum possible time between activation and deactivation. The thrust integrated over that time is the minimum change in momentum (impulse) the thruster can cause, which divided by spacecraft mass provides a minimum possible delta-V. A heavy spacecraft using tiny ion thrusters could achieve very fine precision, but those have only recently become available, and older satellites use thrusters with much coarser control. If the thruster control has some fixed uncertainty, due to, for example, the difference in shape between modeled and actual zero-to-max and max-to-zero thrust profiles, then the smaller the impulse, the greater the relative uncertainty in it: e.g., 100 Newton-seconds (N-s) $\pm$ 1 N-s vs. 1 N-s $\pm$ 1 N-s. On several older geosynchronous vehicles I used to work with, our typical station-keeping burn was about eight seconds, plus or minus a second or so. We predicted the post-maneuver ephemeris from our model of what we meant to tell the thrusters to do, but we always did confirmatory measurement analysis afterwards to determine, based on the observed delta-V, whatever it was that the thrusters actually did.
The need to solve for the real maneuver after the fact seems consistent with this ArianeGroup brochure, which quotes thrust uncertainties of roughly 5 to 10 percent. A possibly useful reference on measuring and modifying the minimum impulse is JPL Tech Report 05-3438, "The Minimum Impulse Thruster", which describes the engineering that went into reducing the minimum impulse of one particular thruster by a factor of five, and the test results that show it worked. For a detailed comparison of different thruster types and the minimum thrust available from each at one point in the not-too-distant past, you might try Johansson, Optimal Thruster Actuation in High Precision Attitude and Orbit Control Systems, Luleå University of Technology thesis, 2005.
Exceeding the north-south (let's call this Y) limit is not generally a problem because there isn't an assigned owner of the space in that direction, partly because that might require your customers to have antennas capable of tracking the spacecraft as it moves across the sky, or suffer significant signal degradation at certain times of day. There's also no way to stay on one side of the Y axis: if you are north of someone else's box right now, then twelve hours later you will be south of it, meaning you must have passed through their box, and will continue to do so twice a day, every day (unless you have a much larger eccentricity than typically used, which would let you trace a diagonal path such that you are above one X neighbor, and then below your other X neighbor, but still cross the equator only in your box). It is possible to have multiple geosynchronous but not geostationary satellites sharing the same slightly inclined and slightly eccentric path (the figure 8 track you've drawn is called an "analemma"), if they're appropriately phased around it to avoid collision, but the more things you pack into the same space, the greater the chance of damage if something goes wrong.
Then, at end of life, you're supposed to enter a "graveyard" orbit, which means precisely that you increase your nadir distance to far enough above the synchronous height that your X and Y drifts don't matter anymore.
