I personally draw a distinction between "balling" a dough and "re-balling" a dough. Balling is what you do with a dough that comes out of a mixer (or a reasonable time thereafter) or a dough that has been bulk fermented, typically at a room temperature or at a controlled temperature, and then divided into pieces and formed into round dough balls. In these cases, the dough is typically soft and malleable enough to be able to fairly easily form into smooth round dough balls. The videos referenced earlier do a good job of showing how to ball a dough. Some members will also use balling or re-balling of a dough after a cold fermentation. This is a method that is usually consigned to a home application although on rare occasion it has been used at a professional level (with the restaurant A16 being one such example with respect to individual dough balls).
To me, the term "re-balling" implies a second or later balling of the same dough. We have some members who do that. John Fazzari is one such member, and he seems to have mastered the technique. And it seems to me that when such a dough is re-balled, it is a fairly gentle action, not an aggressive re-kneading of the dough or using stretch and fold or slap and fold techniques. Using these techniques, especially when the dough is removed from the refrigerator, will invariably alter the gluten structure to the point where the dough becomes highly elastic with a lot of snap-back, making it very difficult to open up to form a skin without rips or tears forming in the skin. To get such a dough to become less elastic, one normally lets the dough relax again so that the gluten relaxes and renders the dough more extensible. However, this can sometimes take several hours (depending on the dough formulation and especially the hydration and fermentation temperature). Sometimes, this palliative does not work especially well even after a prolonged rest. By using a gentle re-balling that more or less tries to re-shape the dough into a round ball by pulling the outer edges of the dough back in toward the center, as shown in some of the videos referenced earlier, and at the same time pressing the dough ball gently to shrink its size, that form of re-balling seems to work for those who, for whatever reason, choose to re-ball their dough balls. Even then, it might be desirable, or even necessary, to let the re-balled dough balls rest for a reasonable period of time before using. That time might be as little as an hour or two but it might be several hours, depending on the dough formulation used and the ambient temperature (usually room temperature).
Because I do not think the term "re-balling" is particularly descriptive of the action taken, my practice when writing about this step is to use a much longer expression. For example, I might caution someone not to "re-ball, re-knead, or otherwise re-work" the dough ball. Re-kneading is a highly descriptive term and needs no further elaboration. Re-working a dough ball is much less descriptive but might include using stretch and folds or similar techniques or slamming the dough against a hard work surface, which is a technique that Nancy Silverton once described where she would hold one of her very young children in one arm as she slammed the dough onto a work surface with her free hand. In her case, she would do this over a hundred times. Now, that is "re-working" the dough