There will be information conflict due to progress in technology and the fact that these bikes are so old now.
When these bikes were designed, mechanical regulators were the norm. They are crude but functional for the battery technology of the time.. Back then, the (lead-acid) batteries were wet-cell and unsealed, so water levels could be maintained by the user. This helped deal with imprecise regulators. If the battery was getting over-charged, the water loss could be replaced by the owner relatively simply. Many/most of us are now using some form of sealed battery which may need a slightly more precise regulation with a voltage level that is likely lower than the permissible range described in the manual. The norm for more modern, fully-electronic regulators was to target the battery voltage to about 14.2 (average voltage). In practice the range, as measured by a digital meter, would be somewhere between 14.2v and 14.5 ish volts. Higher than that, in the summer, might indicate a little maintenance might be needed on the wiring or the regulator might be starting to fail. More modern batteries, especially lithium-based, require even more precise voltage control. As a safety, the permissible limit of the voltage range will probably be lower than that used with lead-acid. This might be where the 13.8v value comes from, but that would be lower than desired for lead-acid batteries. Using 13.8 will cause the battery to re-charge slower.
Also, back then, voltmeters were analog and not very precise. Regulators regulate the *average* voltage. But the "average" can be different for different digital meters that use different sample times. Analog meters show a "continuous" average, but even that will depend on the internal construction and will affect how rapidly the meter will react, so can effect the reading. This can lead to small differences when different meters are used to measure the same battery voltage. This is because the battery voltage is not pure DC. It has a slight pulsing component to it. which is really a small AC component to the overall voltage. Measuring AC or pulsing DC voltage precisely requires "true RMS" meters which are much more expensive than the typical meters the home-mechanic would use. All of this meant that the manual had to be written to take into account the possible variations of volt meters available. I think the manual was written with the expectation that the Kawasaki factory volt meter would be used. I don't think I've ever seen one in person.
As a side note, regarding the manuals out there:
Some manuals have the unloaded AC test voltages flipped. They incorrectly list the 3-phase result as being higher voltage than the 1-phase result. 1-phase should be higher (if I recall, somewhere near 70 or 75 VAC), whereas the 3-phase result should be lower (if I recall, somewhere near 50 or 55 VAC).