How can one reconcile the hadiths that mention that the Prophet did not shake hands with women, and another narration that mentions that a woman took the Prophet by his hand?

It is not permissible for a man to shake hands with or touch a non-mahram (definition) woman, and there is no difference between a young woman and an old woman, except for a very young girl who is not desired, meaning she is under seven years old. The Prophet, peace be upon him, never shook hands with women, nor did his hand ever touch the hand of a non-mahram woman, as he only took their pledge of allegiance through words. As for the hadith (definition) about the slave woman who used to take the hand of the Messenger of Allah, peace be upon him, scholars have interpreted it as an indication of his good manners and his readiness to fulfill her needs, not as an actual physical touching of the hand. This is a well-known Arabic idiom. Moreover, the authentic narrations of the hadith do not mention touching the hand, but merely secluding himself with her to fulfill her need. Any wordings that imply touching the hand are weak and inauthentic.