Some textbooks describe the Zorn Lemma as: Every nonempty ordered set S has the maximal element if every totally ordered subset of S has an upper bound in S. Some other books replace the upper bound with maximal element which requires that the maximal element belongs to the totally ordered subset of S.
Are these two statements equivalent?
On
The version where you require that every chain has a maximum is not the same as the usual Zorn's lemma. It is equivalent to the axiom of dependent choice which is rather weaker than the full axiom of choice.
To show this, consider it in contraposed form: If a partial order has no maximal element, then it contains a chain without a maximal element.
This is implied directly by the axiom of dependent choice, because we can start at any element and then keep choosing a larger element. The set of elements we choose is a chain without a maximum element.
On the other hand, we can prove DC from the weakened Zorn's lemma: Suppose we have a graph where every node has an outgoing edge. Pick a fixed node to start at and consider the partial order of finite paths that start at this node, ordered by the "is a prefix of" relation. This order has no maximal element (every such path can be extended), so by the weakened lemma it has a chain that has no maximal element. The union of that chain must be a ray in the graph, exactly what DC wants to exist.
No, the latter condition is strictly stronger. For example, consider the set $[0, 1]$ ordered by the usual ordering on the reals. Note that $[0, 1]$ has a maximal element $1$ (in fact, the maximum), so indeed every totally ordered subset has $1$ as an upper bound. However, not every totally ordered subset has a maximum. In particular, $(0, 1)$ has no maximum.