Nice poem, Andrew. Congrats! I think it's definitely the same ship in terms of the design and blueprint, but not the same in its materials. It's like us as we age. Are we the same as when we were children? Are we the same person as we were 10 years ago? Each and every cell in our body has been replaced. So are we the same person? Yes and no. But there is something to be said for continuity.
If you could be teleported, and every atom in your body were replaced when you arrived at your destination, you are in essence exactly the same person as you were before you were teleported.
The same goes for the log cabin.
On the other hand ...
https://en.wikipedia.org/wiki/Ship_of_Theseus
The Wikipedia article has several different solutions to the Ship of Theseus paradox. I just read it and found it intriguing.