I will try to answer your question, but I have to make a couple important notes about antenna and transmission line theory first to help you understand it.
1- To get useful power into a half wave radiator, the source impedance must match the feed point impedance, and the feed point impedance depends on the feed point location along that radiator.
2- If the feed point is at the center (quarter wave on both sides), the source needs to have a low impedance
3- If the feed point is at the end of the half wave radiator, the source needs to have a high impedance.
4- Quarter wavelength transmission lines will invert the impedance seen at the other end, so if one end is shorted, the other end is a high impedance. If one end is open, the other end appears as a short.
5- If a shorted quarter wavelength transmission line is used as a series element, it acts like a parallel LC tank circuit shifting the signal by 180°, so the signal applied to one leg will be coming out the other leg 180° out of phase.
Referring to the middle antenna in figure 22.5 on w8ji's web page, the low impedance transmission line is feeding the quarter wave radiator at the top and the quarter wave sleeve "skirt" going down. The two form a half wave dipole. At the bottom of that first skirt, there is a signal with a high impedance (since it is at the end of a half-wave dipole). The inside of that skirt sleeve will act like a parallel LC tank in series with that high impedance shifting the signal by 180°. So what comes out of the inner leg of the sleeve (the signal riding on the coax shield) is a high impedance signal with current traveling in the same direction as the first half wave element. Looking at antenna C, that outer part of the shield acts as the tip of the next half wave element. And since we are driving a half wave element at the end this time, that high impedance signal coming out of the sleeve is perfect. The signal phase matches the top radiator, and the impedance is proper for the driving location.
When we have the desired number of elements, how do we prevent the high-Z signal from the inside of the last sleeve from going down the shield as common-mode current? Again, it comes down to feed point impedance looking into the section you don't want power on. if the combination of coax and radio enclosure is a half wavelength, or multiple, it will gladly accept power from the sleeve above and become part of the antenna. You will have common-mode currents.
Back to your question of which way the sleeve faces. Lets say you put a quarter wavelength sleeve on the shield facing the other way. Any common-mode signal looking into the open end will see a high impedance. Take a look at the next illustration below, section titled "decoupling impedance". In this case, a sleeve with the open end facing the other way, towards the load, prevents common-mode currents. This is because there is a low impedance feed point driving a low impedance load, and since there is a low impedance load, the low impedance signal goes into the load only and the high impedance formed by the sleeve (which is in parallel with the load) has little effect on anything, just like a 10kΩ resistor across an 8Ω speaker terminal. All of the current goes into the matched load
This is often called a bazooka balun. Although mathematically identical to the other sleeves facing down the coax, we are taking advantage of the high impedance to tell those low impedance signals they can't come this way. This only works if there is a mismatch, so the open end of the bazooka needs to be at a low impedance point or it will simply do what the other sleeves did and pass the signal down the coax shield. In figure 22-5 (c), the low impedance point is at the bottom of the antenna where the high impedance coming from the inner part of the sleeve is transformed to a low impedance by a quarter wavelength of shield to where the drawing ends. That would be the ideal location for the open end of the bazooka sleeve.
The bazooka balun can technically be inside the enclosure, but I fear you have missed one of the most important parts of that paper, the choke impedance. The sleeve and the coax shield forms a transmission line and the choking impedance is a direct function of that transmission line impedance, so if the sleeve diameter is too small, your choking impedance will also be small. The ratio of sleeve diameter to shield diameter must be large for any of this to work properly, and a large sleeve will most likely not fit inside your enclosure. If you are going to try this with a sleeve that is tight against the coax jacket, you may as well just abandon the whole idea.
Edit: and don't forget to mind the velocity factor! A tight sleeve with mostly PVC or whatever the jacket is made of filling the gap will have a very different VF from a sleeve full of air (or the outer surface of the sleeve doing the radiating). So remember, if you want this to work, you need a sleeve much larger than the coax diameter, and the space between the sleeve and shield must be mostly air, otherwise the outside of the sleeve (radiator) and the inside of the sleeve (phasing element) will be different lengths electrically (due to the dielectric in the sleeve) and everything goes to shit. Something else worth considering is placing quarter wavelength radials at that low impedance point right above the mouth of the bazooka sleeve so the current at that low impedance point has a place to go instead of seeing just the sleeve and reflecting back up the chain.