Is the Abrahamic prayer sufficient when praying upon the Prophet (peace be upon him) outside of prayer, or is it exclusively for prayer, knowing that praying upon him (peace be upon him) is done by both prayer and salutation together?

There is no harm in sending blessings upon the Prophet, peace and blessings be upon him, outside of prayer using the Abrahamic formula; in fact, it is the best of formulas. Al-Hafiz said: "It is inferred from his teaching, peace and blessings be upon him, to his companions the method—after they asked him about it—that it is the best of ways to send blessings upon him, because he would not choose for himself anything but the most honorable and excellent. And it follows from this: if someone swore to send the best blessings upon him, then the way to fulfill the oath is to do so."

As for the words of Imam An-Nawawi, may Allah have mercy on him, they clarify that he meant sending blessings upon him when his name is mentioned. His intention was that if a Muslim hears the name of the Prophet, peace and blessings be upon him, they should not limit themselves to saying: "O Allah, send blessings upon him," or "Peace be upon him"—rather, they should combine both. It is also commendable for the reciter of Hadith (definition) and others who mention the Messenger of Allah, peace and blessings be upon him, to raise their voice when sending blessings and salutations upon him, without exaggeration.