Do you mean to say that 6500K as a calibration target actually specifies the coordinates of the blackbody 6500K, as a point on the planckian (blackbody) locus?
Yes, that is correct.
The daylight co-ordinates are positioned slightly above the planckian locus in the xy diagram, which means you should expect a slightly higher y value for daylight.
Would you see the difference between D65 and 6500K? Yes and no.
Yes, because it really is a 10 degree observer problem as opposed to the usual 2 degree observer upon which most theory and calculations are based. In addition it is on the neutral axis. We are extremely sensitive to comparative differences in the conditions above.
However, no, because there is no comparative difference. Your screen is either set to one or the other and you will immediately chromatically adapt to the difference. Even if there is something like an absolute perception, as a hypothetical equivalent to absolute hearing, you would not be able to discern the difference temporally.
More over, it is my experience that the stability of the measurement device in combination with the stability of LCD near white is simply not precise enough to measure the differences required to find the exact white point. This may explain why you get better results with one value as opposed to the other. The minute difference between the target whites may give a relatively significant difference in the graybalance after calibration.
With this in mind I would advice the following:
If you have a choice between D65 or 6500K, use the value that gives you the most pleasing, stable, and or comfortable result subjectively.
Have a slight preference for D numbers, because the ICC internals may have fixed values for these. For example, the ICC internals are based on a fixed D50 illuminant. Providing the exact same values for your device white will allow a CMM to skip Chromatic Adaptation calculations.
PS. The "experience" mentioned above is based on the original i1-display and relatively old LCD technology. The results may be different for a quality colori/spectrometer and a state-of-the-art, high-end eizo for example.