You are talking about the well-ordering theorem, not the similarly named well-ordering principle. That’s exactly my confusion when I first opened this PDF.

Different folks use different conventions. When I was taught it, they called it the principle, not theorem. You can find similar comments on the Internet (e.g. math subreddit).

Here's one that acknowledges it:

https://math.stackexchange.com/questions/1837836/well-orderi...

> The "well-ordering principle" has (at least) two different meanings. The first meaning is just another name for the well-ordering theorem. The second meaning is the statement that the usual relation < on the set N is a well-ordering. This statement is equivalent to the statement that ordinary induction on the natural numbers works.