In A Comprehensive Course in Analsysis by Barry Simon (p. 96) (Available in Google Books here.), Simon defines a directed set as a partially ordered set $Z$ with the property that for all $\alpha, \beta \in Z$, there exists $\gamma$ with $\gamma > \alpha, \gamma > \beta$.
The Wikipedia article on directed set (link) defines defines a directed set as a preordered set $Z$ with the property that for all $\alpha, \beta \in Z$, there exists $\gamma$ with $\gamma \geq \alpha, \gamma \geq \beta$.
In trying to reconcile the two definitions I am wondering: Is it the case that any preordered set with a direction (i.e. the property that for all $\alpha, \beta \in Z$, there exists $\gamma$ with $\gamma \geq \alpha, \gamma \geq \beta$) must be antisymmetric and as such automatically a partially ordered set?
No.
Example 1: The words in a dictionary ordered alphabetically but only by first letter is a preordered set (reflexive and transitive) with a direction (since for any two words in the dictionary they will both be $\leq$ "zero"). However, this set is not a partially ordered set since it is not antisymmetric (e.g. "parallel" $\leq$ "poset" and "poset $\leq$ "parallel" despite the fact that "parallel" $\neq$ "poset").
Example 2: The preorder article on Wikipedia gives another example or a preordered set with a direction that is not a partially ordered set. The example given in the diagram that appears there is the natural numbers but ordered by the integer part of $x/4$. That is, $x \leq y$ provided $\left \lfloor{x/4}\right \rfloor \leq \left \lfloor{y/4}\right \rfloor$ (in the article they write this as $x//4 \leq y//4$). Again, this is a preordered set with a direction, however it is not a partially ordered set due to the fact that it fails to be antisymetric (e.g. "5" $\leq$ "6" and "6" $\leq$ "5" despite the fact that "5" $\neq$ "6").