I found this post on another page that talked about the Bender family. The author's name was Prairie Rose.
The Ingalls family was definitely illegally squatting on the Osage Diminished Reserve -- Laura pretty much comes right out and says so in Little House on the Prairie. Pa says, "If some blasted politicians in Washington hadnít sent out word it would be all right to settle here, Iíd never have been three miles over the line into Indian Territory."
Charles Ingalls never filed on a homestead claim in Kansas because the land wasn't yet open to settlers; it was still Osage land. However, Charles and the other settlers mentioned in the book had heard that the land would soon be open, and they knew it would go fast. The best way to be sure of getting a good piece of land was to get there before it was open for settling, and that way as soon as it was available, they could file their claim -- first come, first served, so to speak.
However, it looked like things were going to take a different turn, and the Osage weren't going to be pushed into giving up their land. A meeting was scheduled to discuss what the final outcome would be in August 1870, but it was delayed so the Osage could go on their annual buffalo hunt.
Carrie was born August 3, and the Ingalls family left Indian Territory as soon as Ma could travel, probably late August or early September. However, another piece of information that helps us put together what really happened is that Pa also received word that the man he had sold his farm in Pepin to couldn't pay up.
It's not sure whether Pa truly believed the deal with the Osage had fallen through and he left solely because he thought he had to, or whether they were so worried and upset at the tensions with the Osage that when this information came through Ma thought it was providential and they'd best return, or whether they knew things with the Osage were going to be settled soon in the settlers' favor but with the buyer of the farm defaulting on his payments, they didn't have enough money to sustain living there and were therefore forced to return for financial reasons. Regardless of the reason, the Ingalls family packed up and headed back to their Pepin farm before the September meeting, where the Osage reluctantly signed a treaty forcing them to move to new land, purchased from the Cherokee, thus opening up their territory for settlement by the whites.
As for historical accuracy and what was published in the books, I think it's important to understand that Rose -- in her own works -- cared little for facts and wrote whatever she thought would make the best story. The Little House books aren't purely autobiographical and weren't meant to be -- but they are of course historical fiction, and I believe Laura provided the historical and Rose provided the fiction. :) In some cases, particularly in Little House on the Prairie because she was only 3 when the family left there, Laura didn't have all the facts. While she did do research to try to learn what she could, she was probably forced to do some guesswork when it came to dates and specific locations.