At the Wikipedia page for the well ordering theorem
For every set X, there exists a well-ordering with domain X.
Furthermore, a well ordering is defined to be a strict total order so that each subset has a least element.
Now I have a couple of questions about this:
Does this mean that the set X itself has a least element? After all X is a subset of itself (unless the theorem is talking about strict subsets).
Some books (e.g. Manifolds and Differential Geometry, by J.M. Lee, Appendix B) defines a well ordered set to be one where there exists a partial ordering so that every subset has a least element. And then uses the well ordering theorem together with this definition.
I can easily see that the well ordering theorem implies this (since every total ordering is a partial ordering), but is there any reason why this form is stated and not the "stronger" one directly? Are they perhaps equivalent?
First of all, note that the definition requires that every non-empty subset has a least element, since the empty set is always a subset but it has no least element.
Secondly, yes, it means that $X$ has a least element, at least if $X$ is non-empty. Exactly because it is a subset of itself.
And finally, if you understand "least" as "minimum", then the answer is that the two definition are equivalent, since if $\{x,y\}$ is any two elements subset, then it has a minimum, let's say $x$, so it means that $x<y$. Therefore every two distinct elements are comparable, and the order is a total order.
If you under "least" as "minimal", then the answer is of course negative, since the empty relation is a strict partial order where every element is minimal, so every non-empty set is a minimal element.