I'll try to answer your question, sorry if it doesn't satisfy you

It was mostly the latter. First of all. generally speaking (with exceptions, like in the Muslim kingdoms/communities in the Indian Ocean), the classical concept of a single caliph for the whole Islamic community had had no force since the 13th century with the destruction of Abbasid Baghdad in 1258. There were, to be sure, self-aggrandizing rulers from various parts of the Muslim world including who sometimes included "caliph" among their list of titles. Yet when they did so, it was in a manner virtually interchangeable with "sultan", lacking any implication of descent from earlier caliphs or an exclusive claim to the title or to universal authority. This general disregard for the past prestige of the caliphate seems to have been shared by intellectuals as well as politicians. In the centuries after the fall of Baghdad, Muslim scholars from the extreme traditionalist Ibn Taymiyya to the hyper-rationalist Ibn Khaldun, seem to have reached a basic consensus that the caliphate as an institution no longer had relevance for the times in which they lived.

It was still quite the case when the Ottomans came along. In the first place, There is no evidence to suggest that Selim aspired to the office of caliph before his conquest of Egypt, remaining silent about both in all of his pre-conquest correspondence with the Mamluks. Nor did he make a visible effort to claim these offices immediately following his conquest, failing to make any systematic mention of them in the various fethnāmes sent to announce his victory to officials and dignitaries of his realm, to vassals and foreign rulers, and to his heir, the future Sultan Suleiman. Instead, the title was acclaimed to him by people living entirely outside the territories under his control. The first public acknowledgement of Selim as Khadim al-Haramayn (Custodian of the Holy Mosque, an equally prestigious title) would come four years later from the Sharif of Mecca. Immediately following Selim’s conquest of Egypt, while he was still in Cairo, the Sharif sent an official delegation headed by his own son to meet the sultan and, in a gesture heavy with political significance, to hand over to him the keys to Mecca. It was only after this public act that Selim, in a letter to the ruler of Shirvan, seems to have for the first time referred to the "khilāfet-i ālīye" (exalted caliphate) and to the insertion of his own name in the hutbe of Mecca, reasoning that the failure of the Mamluks to protect the hajj routes had made it incumbent on him to assume this role. Then, in the following year, it was an Indian Ocean Muslim, Malik Ayaz of Gujarat, who as the governor of Diu became the first foreign ruler to spontaneously acknowledge Selim as “Caliph of the Faith” in a letter congratulating him on his victory

Despite its mostly ceremonial title however, The Ottomans did sometimes use it in a politicized manner to mobilize or to correspond with Muslim communities outside the empire. One such example is in the 16th century, when the Ottoman Empire influenced and interacted with Muslim Powers in the Indian Ocean to fight the Portuguese, which i will briefly describe

The treaty of Tordesillas led to King Manuel I of Portugal claiming a new imperial title: "Lord of the Conquest, Navigation and Commerce of Ethiopia, Arabia, Persia and India", as he hoped to be recognized as these rulers’ superior, a "king of kings" or universal emperor, whose authority transcended the physical possession of any specific territory. Because Portuguese ambitions were concerned first and foremost with the transit spice trade, and since trade routes to and from the Red Sea were trafficked principally by Muslim merchants, Portuguese claims would be measured in its ability to prevent Indian Ocean Muslims from travelling to the Red Sea for trade and hajj. For the first time in history a non-Muslim power emerged in the Indian Ocean that was not only capable of preventing disparate Muslim communities from maintaining contact with one another by means of hajj but was actually compelled to do so according to the terms of its own claims to universal sovereignty. The Portuguese then started to raid and blockade parts of the Red Sea, and as a result of this sustained and organized violence, the Muslim Indian Ocean communities was primed for a radical re-politicization of the ideal of the Muslim Umma. With this in mind, the spontaneous acclamation of Selim as both "Caliph" and "Servant of the Two Holy Cities" was a potential two-edged sword. On the one hand, the notion that the Ottoman sultan was now responsible for protecting Muslim merchants and pilgrims throughout the Indian Ocean (and presumably, elsewhere too) implied that his legitimacy could be called into question by events far outside of the empire’s borders, beyond his control, and possibly even unknown to him if he failed to fulfill these obligation. On the other hand, if the sultan were able guarantee the safety of the Indian Ocean hajj, or at least make a credible effort to do so, he might expect a measure of allegiance in return from Muslims throughout maritime Asia, regardless of whether or not they were actually Ottoman subjects. The Ottomans had been nurturing the title caliph since the early reign of Süleyman. These efforts were directly connected with the consolidation of Ottoman rule in Egypt and were spearheaded by the grand vizier Ibrahim Pasha, Süleyman’s closest confidant and a leading proponent of Ottoman military engagement in the Indian Ocean. In 1538, a fleet of over seventy ships eventually did set sail to India from Suez, beginning the history of direct Ottoman military involvement in maritime Asia. With the departure of this fleet, a full-blown revival of the concept of the Universal Caliphate in a thoroughly Ottoman guise began to take shape in the following decades. The clearest evidence for this dates from the 1560s and emerges from a series of diplomatic exchanges between the Ottomans and the Sultan of Aceh, Ali Ala’ad-Din Ri’ayat Syah. In the first letter of this exchange, sent from Aceh in 1564 and addressed to Suleiman the Magnificent, the Ottoman sultan is repeatedly addressed as "caliph" by Ali Ala’ad-Din Ri’ayat Syah and is assured that in this capacity his name is read in the hutbe in all the mosques of Aceh. Moreover, the letter indicates that Suleiman is being similarly named in the hutbes of Sri Lanka, Calicut, and the Maldives, all places strategically located along the maritime trunk routes to the Red Sea, and these communities’ recognition of the sultan as defender of the universal Umma was now expressed in direct expectation of weapons, ships, and technical expertise in return, in order to continue the ongoing fight against the Portuguese

Sources:

Tordesillas and the Ottoman Caliphate: Early Modern Frontiers and the Renaissance of an Ancient Islamic Institution and The Ottoman Age of Exploration by Giancarlo Casale

Legitimizing The Order: The Ottoman Rhetoric of State Power by Hakan T. Karateke and Maurus Reinkowski