\$$\forall x\in X, x \leq x\$$ looks like a standard poset to me, not just a preorder

\$$\forall x,y \in X, x \leq y \$$ and \$$y \leq x \$$ implies \$$x \leq x \$$

Incidentally, have we diverged from Fong & Spivak in these lectures, i.e. do we use poset and preorder with their standard meanings?