There is only one name, which received different subsequent interpretations.
The OD is:
Kuhl H. 1820. Conspectus psittacorum. Nov. Acta Phys.-Med. Acad. Caes. Leop. Carol. Nat. Curios., 10 (1): 1-104.
This name was used in the OD for a section within Psittacus, that included the species: Psittacus guianensis, P. auricapillus, P. squamosus, P. vittatus, P. leucotis, P. versicolor, P. guaruba, P. carolinensis, P. ludovicianus, P. pertinax, P. aureus, P. canicularis, P. aeruginosus, P. viridissimus, P. rufirostris, P. buccalis, P. virescens, P. sosove, P. solstitialis, P. murinus, P. niger, P. rasa, P. mascarinus, P. torquatus, P. alexandri, P. annulatus, P. erythrocephalus, P. barbatulatus, P. benghalensis, P. papuensis, P. pondicerianus, P. xanthosomus, P. haematopus, P. capistratus, P. ornatus, P. lichtensteini, P. lunatus, P. marginatus, P. macrorhynchus, P. grandis, P. ruber, P. unicolor, P. guebensis, P. incarnatus, P. borneus, P. domicella, P. lori, P. guarrulus, P. cyanurus, P. coccineus, P. riciniatus, P. novaeguineae, P. platurus, P. formosus, P. novaezeelandiae, P. ulietanus, P. erythronotus, P. cornutus, P. auriceps, P. concinnus, P. pusillus, P. humeralis, P. discolor, P. australis, P. chlorolepidotus, P. undulatus, P. ultramarinus, P. chrysostomus, P. pulchellus, P. zonarius, P. palmarum, P. spurius, P. venustus, P. cyanomelas, P. erythropterus, P. icterotis, P. eximius, P. multicolor, P. elegans, P. browni, P. scapulatus, P. tabuensis.
When used as valid, it has mainly been treated in a way consistent with a type designation first made by Gray in 1841:
Gray GR. 1841. A list of the genera of birds, with an indication of the typical species of each genus. Second edition, revised, augmented, and accompanied with an index. R and JE Taylor, London.
Here, Gray designated the first species cited by Kuhl, Psittacus "guianensis L." = gujanensis Gmelin 1788 (Linnaeus C, Gmelin JF. 1788. Systema naturæ per regna tria naturæ, secundum classes, ordines, genera, species; cum characteribus, differentiis, synonymis, locis. Edicio decima tertia, aucta, reformata. Tomus primus, pars I. GE Beer, Leipzig.; p. 324;
v. 1, pt. 1 - Caroli a Linné ... Systema naturae per regna tria naturae : - Biodiversity Heritage Library ), a junior synonym of Psittacus leucophthalmus Statius Müller 1776 (now in Aratinga Spix 1824). Gray repeated the same designation in 1855 (Gray GR. 1855. Catalogue of the genera and subgenera of birds contained in the British Museum. British Museum, London.; p. 85:
Catalogue of the genera and subgenera of birds contained in the British Museum. - Biodiversity Heritage Library ). The same type was also accepted, i.a., by Salvadori in 1891 (Salvadori T. 1891. Catalogue of the Psittaci, or parrots, in the collection of the British Museum. Catalogue of the birds in the British Museum. Volume XX. Trustees of the British Museum, London.; p. 170;
Vol 20 (1891) - Catalogue of the Birds in the British Museum. - Biodiversity Heritage Library ).
A year earlier, Gray had designated another species, however, so that his 1841 designation can certainly not be valid:
Gray GR. 1840. A list of the genera of birds, with an indication of the typical species of each genus. R and JE Taylor, London.
Here, Gray had designated Psittacus vittatus Shaw 1812 (Shaw G. 1812. General zoology or systematic natural history. Vol. VIII. Part II. J and A Arch, et al., London.; p. 404;
v.8:pt.1-2 (1812) - General zoology, or Systematic natural history - Biodiversity Heritage Library ), which is preoccupied by P. vittatus Boddaert 1783 (now in Amazona), and a synonym of Psittacus frontalis Vieillot 1818 (now in Pyrrhura Bonaparte 1856).
Last, there is a still earlier designation, by Lesson, that was unearthed by Mathews (Mathews GM. 1911. On some necessary alterations in the nomenclature of birds. Zool. Novitates, 18: 1-22.; p. 11;
v. 18 (1911) - Novitates zoologicae - Biodiversity Heritage Library ):
Lesson RP. 1828. Manuel d'ornithologie, ou description des genres et des principales espèces d'oiseaux. Tome second. Roret, Paris.
Lesson wrote: “Type: le sincialo, psittacus rufirostris, L. enl. 550, et toutes les vraies perruches.”, which Mathews accepted as a designation of Psittacus rufirostris Linnaeus 1758 (Linnaeus C. 1758. Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus I. Laurentius Salvius, Stockholm.; p. 98;
v.1 - Caroli Linnaei...Systema naturae per regna tria naturae - Biodiversity Heritage Library ) as the type of the name (despite the rather odd wording -- a type designation should in principle be the designation of a single species; it may be questionable that the designation of a species "and all the true parakeets" fits the bill). This designation is de facto universally treated as valid in the current systems, given that we use Pyrrhura Bonaparte 1856 for the group to which Conurus Kuhl 1820 would apply if Gray's 1840 designation had been the first one.
Psittacus rufirostris Linnaeus 1758 is not in use, and generally viewed as unidentifiable. But Salvadori 1891 (p. 72;
Vol 20 (1891) - Catalogue of the Birds in the British Museum. - Biodiversity Heritage Library ) had placed Planche Enluminée 550 (
t.6 - Planches enluminées d'histoire naturelle - Biodiversity Heritage Library ), which both Kuhl 1820 and Lesson 1828 had cited as representing P. rufirostris, in the synonymy of "Palaeornis torquata" (his name for the Ring-necked Parakeet), and Mathews 1911 accepted this as making Conurus a synonym of Palaeornis Vigors 1825 (which itself is now a synonym of Psittacula Cuvier 1800).
(Note that, under the current Code, a genus-group name is ultimately typified by the type series of its nominal type species -- other specimens referred to this type species in the OD of the genus-group name play no role. The identity of the type here, if we accept Lesson's designation, must be assessed based on the descriptions of Psittacus rufirostris, as provided by Linnaeus and by the authors he cited (Aldrovandus, Ray, Willughby, Sloane, Edwards, and Marcgrave); what appears on Planche Enluminée 550, which was produced later and not cited by Linnaeus, does not matter, even though that plate was cited by Kuhl and by Lesson. It used to be different under older rules, however, and Mathews certainly thought otherwise.)