Thanks for your your replies.
There are five houses mentioned in the Children's versions of the books: the first two are logs - a log cabin already standing, a log cabin they build in Indian Territory, the other three are built with "bought sawn" lumber. In Plum creek it had a loft, (I think the film was based on this house) and in De Smit a town store and house with a second story, and the homestead claim / homestead that was all on one level, and built in three stages. This last one is what I am trying to replicate. I have found a bit more about these in the following places:
This gives the dimensions of the buildings, but has no contact.
This has lots of tantalising pictures including a replica of the half house shanty they built themselves. Unfortunately they have not updated since 2015, and an email to them came back undeliverable.
This is some information on shanty claims in general, and the one at the top looks like the first pictures of the "half house" shanty in the book.
This has more pictures of the inside of the current building for tourists.
The current homestead people visit, and sketches in later of Wilders book of the homestead show horizontal weather board. Pa talks about tar paper on the house being blown off the roof in a blizzard although well boarded on, and one old photo showed a shack wrapped in tar paper and held on with a mishmash of thin boards.
These may give someone an idea of the name for the style of building I am trying to replicate. Also maybe a tiny house plan I could take and adapt in this style.
I have also learnt that under new laws I can build up to 30m2 without consent provided it is "light" building material, one level, and of no more than 20kg per Square metre wall or roof - e.g. not brick or cement etc. and no electricity, sewerage or potable water.