Norse expeditions around 1000 CE are well established. Polynesian contact is also well regarded at this time, though some people are holding out for an archaeological smoking gun. The linguistic evidence associated with the spread of sweet potatoes through Polynesia is fairly convincing, but an actual archaeological site featuring either Polynesian artifacts in South America or South American artifacts in Polynesia would really lock the theory down. The Nature and Science quotes are actually talking about a different theory, regarding whether the ancestors of the indigenous peoples of Australia and Melanesia also contributed to the initial settling of the Americas back in the Ice Age (and, if they did, whether they crossed the Pacific by boat or came through Beringia like everyone else).

So other than the certainty of Norse contact and the near certainty of Polynesian contact, the rest are highly speculative or entirely refuted by modern anthropology and archaeology.