Zorn's Lemma - partial order, or preorder? According to Wikipedia Zorn's Lemma says;

every partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element.

According to Nlab Zorn's Lemma says something slightly different;

Every preorder in which every sub-total order has an upper bound has a maximal element.

Are these different?  Is the nlab version stronger than the other?
I can see how most of it corresponds;

*

*sub-total order $\mapsto$ chain.

*at least one maximal $\mapsto$ a maximal

but I have to conclude with its weaker preorder condition, the nlab version is a stronger theorem.  Is this correct?
Am I right in thinking that when there are multiple maximal elements, they might look like a cyclic set that fails antisymmtry such $a\preceq b\preceq c\preceq a$, and in this case every distinct element $a,b,$ and $c$ are maximal elements?
I'm unclear whether it can be the case that there are separate distinct maximal chains e.g. $a\preceq a$ and $b\preceq b$ which do not satisfy $a\preceq b$.
 A: There is no essential difference in the sense that one version is stronger than the other.
It is more a question of: one version of the lemma  is translated into another.
Double translations are neutral.

If $(P,\preceq)$ denotes a preorder then it induces on a natural way a partial order $(P',\leq)$.
Elements of $P'$ are equivalence classes wrt to relation:$$x\sim y\iff x\preceq y\wedge y\preceq x$$
Relation $[x]\leq[y]$ on $P'$ defined by $x\preceq y$ appears to be well defined.
A maximal element $[m]$ in $P'$ is actually a set of maximal elements in $P$ and for every $x,y\in[m]$ we have $x\preceq y\wedge y\preceq x$ (so yes, there is cycling there).
On the partial order we have the lemma of Zorn and "translating" it to the original preorder gives a formulation of the lemma for preorders (and vice versa).
IMV it is a good habit to connect any preorder that you meet at once with the partial order that is induced by it.

Edit:
Usually an element $m$ is defined to be maximal if: $$m\leq a\implies m=a\text{ for every }a$$
This works fine in partial orders but not in preorders.
A definition that works for both is:
$$m\text{ is maximal if }m\leq a\implies a\leq m\text{ for every }a$$
