All of which goes to show that there
is a visual aspect to calibration, it's not just a question of dialing in the numbers. The numbers that give you a visual match to paper white are the right numbers - whatever they happen to be. In fact setting the white point could and perhaps
should be done entirely visually.
Often you can only adjust along the blue-yellow Kelvin scale, but not along the green-magenta axis. In that case it may be next to impossible to get it entirely right. This is again an advantage of a high-end integrated hardware calibrator like Colornavigator/Spectraview, where you can set white point freely in two dimensions.
When using a standard software calibrator like i1 Profiler, it may be best to set the white point using the monitor's OSD, and then calibrate to native white. Or even go one step further and set the calibrator to "profile only/no calibration". i1 Profiler doesn't allow this, but Argyll/dispcal does. This will only work with color managed apps of course, but with a good monitor it's a valid option.
The two appear to interpret white point slightly differently
In your case you're using the same sensor, so that's one less variable. But there will always be slight differences between different combinations. On the Eizo CG246 I can see a very slight difference between using the i1 D3 and the built-in sensor. They're equally "accurate"; they just require slightly different white point settings.