Is gesturing in bowing (rukūʿ) and prostration (sujūd) for the incapable done with the head only? And is it sufficient to incline the head, or must the back also be inclined?

The statement that a sick person praying while sitting should only nod their head during bowing (ruku') and prostration (sujud) is incorrect. There are indeed narrations describing the Prophet's supererogatory prayers while on his riding animal, stating that he would nod his head, and prostration was lower than bowing. However, these narrations do not explicitly state the absence of bending the back. Rather, it is possible that the mention of the head was because it is the primary part, with the neck and back being subservient to it in bending. The extent of the nodding is not restricted by the text. Furthermore, these narrations pertain specifically to supererogatory prayers, where the scope for ease is wider than for obligatory prayers. A person praying is required to perform bowing and prostration to the best of their ability, including bending the back, which is what a group of the Salaf and Imams have stated.