The variable resistor is the load for the source/DAC (voltage divider). Its non-linearities directly distort the output of the DAC as evidenced by how the same happens with distortion created in dummy loads for amplifier testing. An active load is just not going to be clean and certainly not for state of the art DACs.
This is correct but not what the plot you posted shows.
It shows integral nonlinerity, that is the deviation from ideal when applying a constant input signal and stepping through the codes which set the point at which the resistor ladder is tapped. Note than in operation the code setting is not changed in the ABX box, just for clarity.
In case of rheostat mode, Fig.39 applies where we can see a constant current is injected and the wiper-to-GND resistance is measured (by measuring the wiper voltage).
In Rheostat mode the distortion with AC signal dominantly comes from the variable wiper resistance vs voltage because the wiper is an array of simple CMOS switches with the typical graph shown in Fig.17:
As we can see, the Rheostat INL exactly tracks the wiper resistance profile (vs. supply voltage) as it should be, because the output voltage varies. At the point where the additional voltage drop on the wiper is largest the INL is off the most in the positive range (higher resistance than should be)
We must note this is a graph taken with an old school method of a stepped signal applied while stepping the codes, so this is actually a re-used analog time-domain plot rather than a correct pin diagram. The input signal is set to a new value every 16 codes, as a part of the measurement procedure, it is not a behavior of the part itself, whose structure looks like this:
The silicon resistor string also has voltage coefficient (and temperature coefficient) but much lower.
All these distortions are relevant when the part actually was used in Rheostat mode which is what educated guessing seems to suggest, looking at the PCB.
In potentiometric mode, the distortion is much lower, notably when not loaded down. The wiper impedance is irrelevant, and the voltage coefficient of the divider with 256 all euqual resistors cancels out. That why in this mode AD states the distortion with 0.014%. Not stellar but good enough I would think to not spoil the test results.
As noted, we don't know which mode is used in the device but if Rheostat mode was used then that certainly would be an unlucky choice.