How do I get the most accurate camera calibration?
You have to take images for calibration from different points of view and angles, with as big difference between angles as possible (all three Euler angles should vary), but so that pattern diameter was still fitting to camera field of view. The more views are you using the better calibration will be. That is needed because during the calibration you detect focal length and  distortion parameters, so to get them by least square method different angles are needed. If you arn't moving camera at all you are not getting new information and calibration is useless.
Be aware, that you usually need only focal length, distortion parameters are usually negligible even for consumer cameras, web cameras and cell phone cameras. If you already know focal length from the camera specification you may not even need calibration.

Distortion coefficient are more present in "special" cameras like wide-angle or 360°.

Here is the Wikipedia entry about calibration.
And here is non-linear distortion, which is negligible for most cameras.