The same name, with much the same character attached to it, is told from Aotearoa to Hawaii and out to Tonga, Tahiti and the Tuamotus, across a spread of ocean that took a thousand years to settle. That distribution is one of the plainest pieces of evidence for a common Polynesian inheritance, and it is used as such.
What travels is not a fixed text. In Aotearoa, Maui-tikitiki-a-Taranga fishes up the island and dies attempting to pass backwards through the body of Hine-nui-te-po, and with his failure death becomes permanent for everyone after him. In Hawaii, as Martha Beckwith's survey of Hawaiian mythology sets out, the fishing and the snaring of the sun are fixed to particular named places and the death narrative is largely absent. His standing shifts with the telling as well: in some island traditions he is a being of exceptional power reckoned within the human line rather than an atua receiving cult, and in others he stands a good deal closer to a god than to a man.
Most of what reached English came through two collections. George Grey published his Maori material in 1855, taken from named narrators, and it was that book which carried the figure into English language reading. Beckwith's survey did the same work later for the Hawaiian traditions.
Reading these as versions of one story to be reconstructed misses the point, and reading them as a single pan-Polynesian myth flattens traditions that belong to particular islands and particular lineages. Each account is the property of the people who tell it.