Is Caelus the Roman equivalent of Uranus, or did the Romans adopt the latter and incorporated him in their mythology while keeping his original name?
I've seen multiple conflicting answers on the subject. [The French Wikipedia page for Uranus][1] claims it is a Roman god, while the English translation of the same article points to Caelus.
Besides, it is often claimed that Uranus is the only planet named after a Greek god. (Calculating the Cosmos by Ian Stewart).
I can find no evidence that Uranus was a Roman god. Note that the article you cited in your question provides no sources.
In academic sources, I can find no mention of Uranus as a Roman god. Among the sources consulted are:
Mary Beard, John North & Simon Price, 'Religions of Rome' (2 vols) (1998)
Michael Lipka, 'Roman Gods' (2009)
Malcolm Couch, 'Greek & Roman mythology' (1997)
D. M. Field, 'Greek and Roman mythology' (1977)
Further, Uranus does not appear on this List of Roman deities.
What is perhaps somewhat confusing is that Uranus is the Latin spelling of the Greek Οὐρανός (Ouranos), but this does not make him a Roman god (though the Romans knew of him). Also confusing is the naming of the planet Uranus by Johann Elert Bode. On the naming of the planet:
In a March 1782 treatise, Bode proposed Uranus, the Latinised version of the Greek god of the sky, Ouranos. Bode argued that the name should follow the mythology so as not to stand out as different from the other planets, and that Uranus was an appropriate name as the father of the first generation of the Titans.[46] He also noted that elegance of the name in that just as Saturn was the father of Jupiter, the new planet should be named after the father of Saturn.
The problem here is that, in Roman mythology, Saturn's father is actually Caelus but Bode was an astronomer, not an expert on mythology.
Of course, Uranos is basically Caelus: a Greek god as seen in the interpretatio Romana a mere translation; that is a god Romans recognised, but themselves did not worship very much, as already in classical Greek time no cult for him survived:
(Οὐρανός, Lat. Uranus). Divine personification of the sky, treated by Hesiodus (Theog. 126-128) as a mythical figure. U. is born of Gaia, the earth, without the contribution of a father, 'so that he may wrap her up and the gods have a permanent seat in him' (ibid.). After that, U. begets the Uraniones with Gaia (ibid. 424; 486), namely the Titans, including Kronos, Zeus' father. As a result, U. is the progenitor of the gods (ibid. 44 f.; 105 f.). The Cyclopes and the Hekatoncheires (ibid. 132-152) are descended from him as well. After U. locks up the Titans inside the earth, Gaia persuaded her son Kronos to take revenge. When U. wants once again to mate with the earth, Kronos cuts off his genitals with a sickle and throws them into the sea. From the drops of blood falling on the earth, the Erinyes, Giants and Nymphs are born, while the genitals floating in the sea and their foaming sperm produce Aphrodite (ibid. 173-206).
This earliest Greek myth of the creation of the world originates from the myth of the separation of sky and earth, which is spread in many cultures [1]. In his role as Gaia's spouse, U. embodies the sky's generating power, which permeates the earth with warmth and humidity (cf. Aesch. Danaides, TrGF 3 F 44). Hesiodus' story, especially the succession-line U. - Kronos - Zeus goes back, quite obviously, to Oriental models. Thus in Hurrian-Hittite texts, after Alalu (who has no Greek counterpart), the sky Anu (= Sumerian An) corresponds to U.; Anu is castrated by Kumarbi (= Kronos), gods are born out of his genitals; Kumarbi devours his offspring (and a stone?), with the exception of the weather god (= Zeus), who in turn overthrows him. In Akkadian texts, especially in the Enūma elı̄š , the parallelism is not so evident, Apsȗ and Tiamat, however, make up a primordial couple similar to U. and Gaia. In the Phoenician story of Sanchuniathon, as in the Hittite one, there are four generations: Elium (Alalu), Epigeus (U., Anu), El (Kronos, Kumarbi) and Demarus (Zeus, weather god) [2. 18-31].
Uranos was nowhere worshipped with a cult and no figurative representation ever developed.
— Lutz Käppel: "Uranus", in: Brill’s New Pauly, Antiquity volumes edited by: Hubert Cancik et al., 2006 doi
Compared to
Translation of the Greek Uranus (‘Heaven’). The genealogy of C. (Cic. Nat. D. 2,63.3,44; Hyg. Fab. praef. 2) corresponds with some variations to that in Hesiod. Varro (Ling. 5,57) named C. and Terra as the oldest of the deities. C. had no cult in Rome; inscriptions venerating him as aeternus (CIL VI 181-84; cf. also Vitr. 1,2,5) refer to foreign cults [1]. Graphically, C. is portrayed as a bearded man holding a garment above his head in the shape of an arch, as for example on the breast plate of the statue of Augustus of Prima Porta [2].
— René Bloch: "Caelus, Caelum, in: Brill’s New Pauly, Antiquity volumes edited by: Hubert Cancik et al., 2006. doi
There seem to be some complications over interpreting the term "a Roman god" in this case?
While Uranus may indeed not be a native Roman god — and in the translated form of Caelus things get ever more complicated, since that word caelum is also just the plain 'sky', exactly like ouranos — we might refer to Roman authors to clear up these deliberations:
'Well now,' he would say, 'if these brothers are included among the gods, can we deny the divinity of their father Saturn, who is held in the highest reverence by the common people in the west? And if he is a god, we must also admit that his father Caelus is a god. And if so, the parents of Caelus, the Aether and the Day, must be held to be gods, and their brothers and sisters, […]
— Cicero "De Natura Deorum" (3,44), from the translation in "Cicero In Twenty-Eight Volumes XIX, De Natura Deorum. Academica. With an English Translation by H. Rackham, M.A.", Lloeb Classical Library, Harvard University Press: Cambridge, 1967.
Perhaps enlightening for the astronomical terminology angle are these two takes from Wikipedia:
The discovery of a new planet started a debate about its naming that lasted more than sixty years. Herschel himself named it Georgium Sidus - George's Star, in honor of the English King George III. The Jesuit and astronomer Maximilian Hell had suggested Urania, the name of the Muse of Astronomy. In France, astronomers called him Herschel until Bode suggested naming him after the Greek god Uranos. However, the name did not prevail until about 1850 and was adapted to the Latin spelling, corresponding to the Roman names of the other planets. In Roman mythology Uranus is the father of Saturn, who in turn is the father of Jupiter.
— Wikipedia: Uranus (Planet (translated from German)
And:
Uranos (ancient Greek Οὐρανός Ouranós, German: 'Himmelsgewölbe', Latin Uranus, Coelus or Caelum).
— Wikipedia: Uranos (translated from German)