Why do small objects have to preverse colimits from directed sets, rather than colimits from semilattices? Let $\kappa$ be a regular cardinal. Let $\mathcal C$ be a category. Let $A : \mathcal C$ be an object of $\mathcal C$. As best I can tell, we say that $A$ is a small object of $\mathcal C$ if, for all $\kappa$-directed preorders $\mathcal J$ and diagrams $D : \mathcal J \to \mathcal C$ such that $D$ has a colimit in $\mathcal C$, we have an isomorphism between $\mathrm{colim}_X\ \mathrm{Hom}_\mathcal{C}(A, D(X))$ and $\mathrm{Hom}_\mathcal{C}(A, \mathrm{colim}_X D(X))$. (It perhaps makes sense to demand that $\mathcal C$ has all such limits, I can't tell from the nlab page whether that's standard.)
Now, define a $\kappa$-semilattice to be a preorder equipped with least upper bounds for any $\kappa$-indexed set of objects. (Contrast with the definition of a $\kappa$-directed preorder, which is equipped with mere upper bounds for any $\kappa$-indexed set of objects.)
Question: Why does the definition of a small object quantify across $\kappa$-directed preorders, instead of $\kappa$-semilattices? Is it just because $\kappa$-directions are a bit more general? Is it because it makes some of the small-$\kappa$ edge-cases work out in some desired fashion? Is it a mere convenience?
In particular, my intuitions say that if we ignore some of the small-$\kappa$ edge-cases, then the two definitions should work out to basically the same thing, but I'm not sure if I'm failing to visualize a way in which something goes terribly wrong if you use the semilattice version and $\kappa$ is particularly large.
(Note that by "small-$\kappa$" I mean 0, 1, and 2, which we may or may not consider regular cardinals depending on our choice of definition.)
 A: If $J$ is a directed preorder, let $\bar J$ be the free cocompletion of $J$ under finite joins. There is a canonical functor $J\to \bar J$ which is cofinal, since a cocone under a diagram $F:\bar J\to \mathcal C$ is canonically determined by its restriction to $J$. Meanwhile every functor $J\to \mathcal C$ extends canonically to $\bar J$, assuming $\mathcal C$ has finite coproducts. Thus, assuming $\mathcal C$ has finite coproducts, then it has directed colimits if and only if it has colimits of diagrams indexed by sup-semilattices, and we have also the analogous claim for functors out of $\mathcal C$ preserving such colimits. Thus in a $\mathcal C$ with finite coproducts, an object is $\aleph_0$-small if and only if its homs out commute with colimits indexed by sup-semilattices. The generalizations to uncountable cardinals hold as well. 
However, it is not unusual that we want to consider directed colimits in a category $\mathcal C$ lacking finite coproducts-this is more or less the subject of finitely accessible categories. In this case it is not clear that we can extend a $J$-indexed diagram to $\bar J$ as we needed above. That's no problem for $\kappa=\aleph_0$-in fact, a category has and a functor preserved directed colimits if and only if it preserves colimits of chains, that is, functors indexed by ordinals. As totally ordered sets, chains certainly have finite coproducts. This result is 1.7 in Adamek and Rosicky's book on locally presentable categories.
However, it is not the case that general $\kappa$-directed colimits can be constructed from colimits of $\kappa$-filtered chains (which would be the kind of chains forming $\kappa$-semilattices in your sense.) So it seems conceivable that there may exist, say, a category (necessarily lacking $\kappa$-small coproducts) in which there is some object homs out of which commute with colimits over $\aleph_1$-semilattices but not over all $\aleph_1$-directed preorders. It is easy to imagine that whether such a category and object exist is an open question, but I don't know for sure. I also agree with Mark that the canonical diagram shouldn't be a $\kappa$-semilattice in general in an accessible category, although it is in a locally presentable category and in the main examples of non-presentable accessible categories that come to mind. 
So in summary, nothing goes wrong for $\aleph_0$-small objects, but something might go wrong for larger $\kappa$; it seems at best not obvious whether this is the case. That, together with the fact that focusing on $\kappa$-semilattices doesn't give any immediate simplification to the theory, seems sufficient to justify sticking with $\kappa$-directed sets in general.
By the way, a $\kappa$-directed preorder should have upper bounds for subset of cardinality less than $\kappa$, so you probably want to define a $\kappa$-sup-semilattice analogously. For instance an $\aleph_0$-sup-semilattice is then just a sup-semilattice.
A: Let's first see why we would call such objects small.
Concrete example: let $\kappa = \omega$ (the countable cardinal) and work in the category $\mathbf{Set}$ of sets. Then a set $A$ is $\omega$-presentable ("$\omega$-compact" on the page you linked) precisely when $|A| < \omega$, i.e. precisely when $A$ is finite.
To see this, note that an isomorphism $\operatorname{colim}_X \operatorname{Hom}(A, D(X)) \cong \operatorname{Hom}(A, \operatorname{colim}_X D(X))$, for an $\omega$-directed diagram $D$, means precisely that any arrow $A \to \operatorname{colim}_X D(X)$ will factor through the diagram in an essentially unique way. That is, it factors as $A \xrightarrow{f_Y} D(Y) \to \operatorname{colim}_X D(X)$ where $Y$ is some object in our $\omega$-directed preorder. The "essentially unique" requirement means that if $f$ factors through both $f_Y: A \to D(Y)$ and $f_{Y'}: A \to D(Y')$, then there is $Y, Y' \leq Z$ in the preorder such that $D(Y \leq Z) f_Y = D(Y' \leq Z) f_{Y'}$.
Now suppose $A$ is finite and let $f: A \to \operatorname{colim}_X D(X)$ be a function. We have to show that it factors through the diagram. Since $A$ is finite, we can write $A = \{a_1, \ldots, a_n\}$. For each $1 \leq i \leq n$ there is $X_i$ such that $f(a_i) \in D(X_i)$. This is because a colimit in $\mathbf{Set}$ is just (a quotient of) a union. Since the diagram is $\omega$-directed, there must be some $Y$ that is the upper bound of $\{X_1, \ldots, X_n\}$. Hence $f$ factors through $D(Y)$.
Conversely, suppose $A$ is $\omega$-presentable. Let $\mathcal{P}_\text{fin}(A)$ be the set of finite subsets of $A$. This becomes an $\omega$-directed diagram when we consider the inclusions between those subsets. Since clearly $A = \operatorname{colim} \mathcal{P}_\text{fin}(A)$, we must have that the identity $Id_A$ factor through the diagram. But then it factor through a finite set, so $A$ must be finite.

So let's look at the proof strategy. In the forward direction ($A$ finite implies $A$ $\omega$-presentable) we had an arrow $f$ into some $\omega$-directed colimit. We were able to piece together the image of $A$ under $f$ already in the diagram itself, and that is precisely what we want. We want to say that $A$ contains so little information, that if we send that information into a colimit (think: "a union"), then all bits of information can already be pieced together inside the diagram of that colimit. This is the intuition why we do not require least upper bounds in the diagram.

These kinds of objects play a central role in locally presentable categories and the more general accessible categories (where indeed the existence of $\kappa$-directed colimits is assumed for some big enough $\kappa$). One important property there is that every object can be built as a $\kappa$-directed colimit of $\kappa$-presentable objects (much like we did with $A = \operatorname{colim} \mathcal{P}_\text{fin}(A)$ before). In general these diagrams will not be $\kappa$-semilattices.
