The claim of papal supremacy held that the pope had authority over all religions on earth. the pope was the chief authority over the papal states. the pope had authority over all kings and emperors. the pope was the head of the western christian church.

Best  "The pope was the head of the Western Christian Church."

It is also true that popes sought to exercise their authority over kings and emperors as a part of their claim to supremacy.  But the main definition of "papal supremacy" has to do with the pope's position as head over the universal (aka "catholic") church.  The Catechism of the Catholic Church (paragraph 882) sums up the definition of papal supremacy in these words:  "The Roman Pontiff, by reason of his office as Vicar of Christ, and as pastor of the entire Church has full, supreme, and universal power over the whole Church, a power which he can always exercise unhindered."