What is the medieval Muslim title often used for rulers like Saladin?

The story behind the answer

Saladin ruled Egypt and Syria with a title that signified independent, sovereign authority over a realm, while still acknowledging the spiritual primacy of the Abbasid caliph. Rising to prominence under the Ayyubid dynasty, he used this office to unify much of the eastern Mediterranean during the Crusades. The term had gained prestige earlier with the Seljuks, marking a powerful ruler whose legitimacy was strengthened by recognition from the caliph.

This role differed from other Islamic titles: a caliph was the supreme religious-political leader of the ummah; an emir was a commander or provincial prince; and a vizier served as a chief minister. Saladin’s coinage and Friday sermons were proclaimed in the caliph’s name, even as he exercised practical control—most famously reclaiming Jerusalem in 1187.