As many other German surnames, the name Weizsäcker has been around for centuries, has been written in numerous different ways over time and the name’s origins are somewhat clouded.
According to the German version of the Wikipedia article linked above, the family themself trace their roots to a relative of the 13th century medieval knight Peter Wazach or Wadtsacher which inhabited the Waitzacker farm near Weilheim in Oberbayern. Waitzacker seems a plausible alternative spelling since the two are pronounced similarly and some exchanges between t, z and s are always possible. Waitzacker fits Bavarian phonology better than Weizsäcker due to the a sound replacing the ä sound. This theory would correspond to your second attempt (there is no mill at or near the Waitzacker farm but plenty of fields).
Documents can only trace the family back to the 16th century to a miller called Friedrich Weidsäcker born 1535. It seems to be plausible according to the source Wikipedia cites that this person traces back to a miller of the Waadsacker Mühle, today called Woogsacker Mühle. Coincidentally, this mill was in the possession of the above mentioned knight Peter Wazach.
The family of the Weizsäckers continued to be millers for centuries; but one branch — known as the Öhringen branch from their hometown — managed to become the cook (Hofmundkoch) of the Count of Hohenlohe-Öhringen which initiated their social rise into the burgeoisie. Carl Heinrich Weizsäcker was the first member of the family to be promoted into aristocracy (albeit his title was not hereditary) in 1861. His son then later managed to acquire a hereditary title in 1916 (only just in time before aristocracy was abolished in 1918) allowing him to carry the particle von in his name.
After that bit of family history, let’s go back and take another look at the name. Above I have noted how Weiz + Acker is one plausible origin. But what about Weiz(en) + Säcker? Well, I have no grounds to dismiss it. As I said before, names are complicated and name origins are clouded. But with the family history being as it is, especially with the different name variants of Waadsacker, Waadsecher, Waidsacher, Waidsecker, Weidtseckher, Woodsacker and Wadsacker it sounds just a little less plausible.