How sound is it to apply the term "Divine Insurance" to the mentioned hadiths that speak of Allah's preservation of the Muslim, and is it permissible to name these hadiths with this name?

The evidence confirms what the questioner mentioned regarding the supplications and remembrances (adhkar) that encourage their recitation and regular observance. For instance, whoever prays Fajr is under the protection of Allah. Whoever recites Ayat al-Kursy after every obligatory prayer will not be prevented from entering Paradise except by death. Whoever says: "In the Name of Allah, with Whose Name nothing on earth or in the heavens can cause harm, and He is the All-Hearing, the All-Knowing" three times, nothing will harm him. The virtue of Sayyid al-Istighfar (the master of seeking forgiveness) has also been affirmed. It is also established that the Prophet, peace and blessings be upon him, used to entrust some of his companions to Allah Almighty, saying: "I entrust you to Allah, with Whom nothing is lost." It is also affirmed that he, peace and blessings be upon him, said: "Indeed, Allah, the Most High, if something is entrusted to Him, He preserves it." As long as the mentioned proofs for these protections are sound, there is no harm in calling to remembrances and righteous deeds in this manner. The phrase "Divine Security" (Ta'min Rabbani) has been used by some prominent contemporary scholars, including the esteemed Sheikh Al-Sha'rawi and Dr. Al-Zuhayli, and no one has opposed it. Al-Mu'min (The Giver of Security) is the One who confirms His messengers by showing their miracles, and who confirms to the believers what He has promised them of reward, and who confirms to the disbelievers what He has threatened them of punishment. He is the One who secures His allies from His torment and secures His servants from His injustice.