The version of Zorn's lemma that I have found more often is
Zorn's Lemma (1) If every chain belonging to the partially ordered set $S$ has an upper bound in $S$ then $S$ contains a maximal element.
I read the following version of Zorn's lemma, which I find somewhat seemingly different from (1):
Zorn's lemma (2) In any ordered set where every chain has a supremum in the set, any element $b$ has a maximal element $m$ such that $m\ge b$.
where a chain is a totally ordered subset and the supremum is defined as the least upper bound. As to the ordered set, the book does not define what it means to be ordered, but I know that many authors define an ordered set as a set where a partial ordering is defined, although I am not sure that a total ordering is not what is intended here. My translation is literal. This version of Zorn's lemma is said to be equivalent to in an ordered set every chain is included in a maximal chain (with respect to inclusion), and I know that this last lemma, if we mean partially ordered by ordered, is the "usual" Hausdorff maximal principle, which I know to be equivalent to (1).
In particular I am not sure that we can substitute the upper bound with the supremum and the maximal element with $m$. Moreover, I suppose, although the wording of the book is not very clear to me, that any element $b$ has a maximal element $m$ such that $m\ge b$ means that there exists a $m$ in the ordered set h that, for all $b$ in the ordered set, $m\ge b$; but if the ordering is not total, I do not think that we generally can compare the maximal element of the "usual version" of Zorn's lemma with all $b\in S$.
Are such two versions of Zorn's lemma equivalent and, if they are, how can it be seen? I heartily thank you for any answer!
$^1$V. Manca, Logica matematica, an Italian book of introductory mathematical logic and theoretical informatics.