It seems pretty ubiquitous that whenever Greenland or Iceland are brought up people will mention that they swapped the names between the two islands as some sort of defense and tourism strategy to encourage people to go to Greenland and stay away from Iceland? How true is this? Do we have any primary source documents from the naming of the countries that would say otherwise? Do we know of any names the islands would have been called previously? Is the story even backed up by the Vikings language calling them the same thing?
Not only are the names the same in the Norse language, the story behind both names is in the Icelandic sagas. As far as we know, these are the only names they've ever had.
And Greenland, according to Erik The Reds' saga, was named by its discoverer to convince people to move there to live. He wanted to form a colony there, and oversold it to the Icelanders. However, that doesn't really have anything to do with name of Iceland, which was settled a century before.
According to the sagas, Iceland was named as such because the first attempt at settlement failed due to the harsh winters, and it's worth remembering that Iceland does actually have massive glaciers and harsh winters: even compared to Norway it is icy.
Iceland had possibly been visited by irish monks, the book of Icelanders claims they fled when the first Scandinavians arrived. They would not have called it Iceland, but maybe called it by a mythical northern country called Thule.
What is always essential to remember when it comes to Scandinavians in the Viking age however, is that while they could write, they didn't write down the kind of historical record that would tell us for certain why the islands were named as they were. The Norse handed their history down through the generations orally, and they were only written down centuries after the stories they told. Because of that, they are not very reliable as sources to what happened at the time they describe. The stories behind the names seem likely, but they might exist exactly because of that. Just like we today think Greenland has a weird name, the Icelanders of the 13th century would probably have thought the same, and are likely to have asked why it had that name, and it is possible that the story originated as a logical conclusion. However, it is the best explanation we have.
The episode OP has heard is probably the following one, found in the chapter 6 of the Book of the Icelanders, the oldest indigenous (local) writing in the early 12th century Iceland:
'The country called Greenland (Grönland) was discovered and settled from Iceland. A man from Breiðarfjörðr called Eiríkr the Red went out there from here, and took possession of land in a place that has since been called Eiríksfjörðr. He gave a name to the country and called it Greenland, and said that it would encourage people to go there that the country had a good name (Grønlie trans. 2006: 7).
Thus, though not tourism/ sightseeing itself, the episode in medieval narrative tell us the basic framework of OP's story that the current place name had been granted to attract more attention of to-be-settlers to this new land.
On the other hand, a foreign but older source, Adam of Bremen in late 11th century Germany, tells us another possible origin of this place name. He states that:
'In the ocean there are very many other islands of which not the least is Greenland, situated far out in the ocean.......The people there are greenish from the salt water (caerulei a salo), whence, too, that region gets its name.......(Adam, IV-37 (36), Tschan trans. 2002: 218)'.
We cannot have any other contemporary source to verify which origin story is more likely to have been true, but at least the place name of Greenland (and Iceland) as well have been of Old Norse vernacular origin, and both stories must have been transmitted from the Middle Ages, ca. at most three or four generations after the initial 'discovery' of the island alleged by Eiríkr the Red. Neither of them were necessarily based on the real story behind the place name, however.
Adam of Bremen also identifies Thule, an island in the North Atlantic found by Greek adventurer, Pytheas of Marseilles (ca. BCE 4th century), indeed as Iceland (Adam, IV-36 (35), Tschan trans. 2002: 216f.). Some other early medieval authors, including Bede the Venerable (d. 730), mention Thule in their works, but we don't know whether Thule was really Iceland or whether all the Thule in their works was the same island that Pytheas had found.
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I can only do the Iceland part.
The PR bit is a myth. The best source for the name of Iceland is that it's the name given by explorer Flóki Vilgerðarson (commonly known as Hrafna-Flóki) after his visit prior to settlement of Iceland.
The best source we have for the naming of Iceland is from Landnáma, a 12-13th century book that touches on the settlement of Iceland. Obviously the book is written at least centuries after the actual settlement, so it's up for debate how accurate it is.
According to Landnáma, Iceland was initially named Snæland by a viking called Naddoddur. The name literally translates to "Snowland". The second expedition was spearheaded by a man named Garðar Svavarsson from Sweden who gave the land the name "Garðarshólmi" (Garðar's Island).
The third expedition to be mentioned is Flóki's expedition. He overwintered in Iceland and suffered the rough winter. After hiking up a mountain in the western part of the country he saw drift ice in a fjord. He then named the land Ísland (Iceland).
Here's the text of Landnáma (in Icelandic) https://www.snerpa.is/net/snorri/landnama.htm