How can the Prophet's insistence on carrying out the punishment for theft on the Makhzumiyyah woman, and his refusal to be lenient in it, be reconciled with what scholars and shaykhs have mentioned about the Prophet's unwillingness to carry out the punishment for adultery on Ma'iz and Al-Ghamidiyyah?

There is no contradiction between the incidents of Ma'iz and the Makhzumiyah woman, for each has different rulings. The punishment for Ma'iz was established by his confession, and the Imam has the choice to either disregard it or impose the punishment. The punishment for the Makhzumiyah woman was established by clear evidence, and in this case, intercession is not permissible once the matter reaches the ruler.

It is recommended to forgive (or overlook) punishments among people before they reach the ruler. However, if the matter reaches the ruler and is established by clear evidence, then it is obligatory to implement the punishment, and intercession therein becomes forbidden.