As you can see from the quotes you've given, it was quite common to refer to non-Christians as knights. A knight was a social status, and although piety was rather important to knights, knighthood also entailed virtues and behaviours that were not religiously exclusive. Christians, especially those engaged with crusading, recognised that their enemy shared many of the same ideals as they did. Like the western knights, Muslim noblemen were expected to be keen equestrians as well as pious, generous, and loyal men. In the Islamic world, the term 'fáris' encapsulated these qualities, and although chevalier and faris were not the same (behaviour toward women is more important in European chivalry compared to in the Muslim world, and poetry was vital in Muslim nobility but generally not in chivalry), there was a large area of overlap that was apparent to both groups.

This was apparent to the crusaders right from the First Crusade. After the Battle of Dorelaeum, the author of the Gesta Francorum - the main source for the First Crusade - wrote that:

Who will ever be wise or learned enough to dare to describe the prudence, prowess, and valour of the Turks? They believed they could terrify the Frankish race by threatening them with their arrows, as they had terrified the Arabs, Saracens, Armenians, Syrians, and Greeks. But, please God, they will never be as powerful as our men. Indeed, the Turks say that they are related to the Franks and that no man ought by nature to be a knight save the Franks and themselves. I speak the truth, which no one can deny, that if they had always been steadfast in Christ's faith and in Christianity, if they had wished to confess one true Lord, and if they had honestly believed in good faith that the Son of God was born of the Virgin, that he suffered and rose from the dead and ascended into heaven in the presence of his disciples, that he has sent the perfect comfort of the Holy Spirit, and that he reigns in heaven and on earth; if they had believed all this, it would have been impossible to find a people more powerful, more courageous, or more skilled in the art of war. By the grace of God, however, we defeated them.

But the best example of this acknowledgement of shared ideals is in Saladin. After he took Jerusalem in 1187, the sultan was swiftly demonised by Christian authors and denounced as an agent of Satan and lieutenant of the Antichrist, etc. However, when the participants of the Third Crusade actually met Saladin, they quickly realised that - other than his religion - he was pretty much the ideal man by chivalric standards. He was generous beyond measure, merciful, kind, and a cunning warrior. They even begrudgingly accepted that he was a deeply pious man, even if they thought he was worshipping the wrong god. The poet Ambroise, composing an epic of the Third Crusade, said of the Muslim army 'No more valiant men were to be seen, had they not been unbelievers' and of Saladin's brother Saphadin he says 'It was the noble Saphadin d'Arcade, who had committed generous, kind and noble deeds'. The poem Order of Chivalry, written in the early 13th century, is the best example. The poem is framed as a conversation between Saladin and a captive noblemen, Hugh II of Tiberias. Saladin, curious about European knights, asks Hugh to knight him. Hugh is disgusted at the idea of knighting a non-Christian, at which point the two launch into a discussion of what makes someone chivalric. Over the course of the poem, they find more common ground and Hugh comes to the realisation that Saladin would make an outstanding knight, which culminates in Hugh knighting Saladin except for parts of the knighting ceremony that involve touching Saladin with a sword. Hugh's refusal is not on religious grounds, however, but because Hugh believes that only someone of greater nobility and rank can perform the full ceremony and considered himself inferior to Saladin. The poem is an explicit statement and argument that chivalry transcends religion, and a scathing rebuttal of Saladin's portrayal by the papacy. The prevailing view after the Third Crusade (and to a lesser extent before it) was that individual Saracens could be chivalric, but that Saracens as a group remained evil despite these qualities.

The idea of the 'chivalric Saracen' was not just an accepted notion, but became something of a trope in crusading accounts and poems. Sometimes this was done just to flesh out the antagonist and give the hero a worthy opponent, but this trope was also used to make a pointed observation that the idea of nobility and chivalry was not exclusive to one religion, particularly after the Third Crusade with Saladin and Saphadin being commended across accounts of the crusade. The Fourth Crusade only helped this image, as the Sack of Constantinople stood in stark contrast to the mercy shown toward Jerusalem by Saladin in 1187.

Sources:

Ailes, Marianne. "Chivalry and conversion: The Chivalrous Saracen in the old French epics Fierabras and Otinel." Al-Masāq 9.1 (1996): 1-21.

Irwin, Robert. "Saladin and the third crusade: A case study in historiography and the historical novel." Companion to historiography (1997): 139-52.

Keen, Maurice. Chivalry. Yale University Press, 2005.