Why did the Prophet (peace be upon him) name his two sons Ibrahim and Al-Qasim, and not Abdul Rahman or Mahmoud, despite his recommendation: "The best names are those that are praised and those that indicate servitude [to Allah]"?

There is no basis for the hadith: "The best names are those that begin with 'Abd' (servant of) and 'Hamd' (praise)." The most beloved names to Allah are 'Abdullah (Servant of Allah) and 'Abdur-Rahman (Servant of the Most Merciful), but this does not preclude naming with other names. The Prophet, peace and blessings be upon him, named his son Ibrahim, and he also named the son of Abu Musa, Ibrahim. He also named the son of Abdullah ibn Salam, Yusuf. This indicates the permissibility of naming with the good names of the prophets, for the Sunnah is taken from his sayings, actions, and approvals, peace and blessings be upon him.