How authentic is the Hadith: "And he (peace and blessings of Allah be upon him) used to invoke blessings upon himself in the first tashahhud and other instances," and what is its meaning?

The aforementioned Hadith was not reported in Sunan Al-Tirmidhi, and the Prophet's prayer upon himself in the first Tashahhud has not been established. Scholars have differed on this matter: Imam Al-Shafi'i, in his new opinion, deemed it desirable, whereas Imam Ahmad, Imam Abu Hanifa, Imam Malik, and others did not consider it desirable. This latter view is also Al-Shafi'i's old opinion. Those who advocate for it cite evidence such as the general command to pray upon the Prophet, and that this is a place where greeting him has been legislated. Others believe that the first Tashahhud is not the place for it, as a form of ease, and that the Prophet's action of doing so in it has not been established. Furthermore, they argue that the Hadiths used by those who advocate for it are weak and do not support this claim.