A category which direct limits but no general colimits

Consider the category with two objects and only identity arrows. Or more generally, any poset which has least upper bounds for all chains, but not arbitrary joins (like the disjoint union of two copies of $\mathbb{R}\cup\{\infty\}$).

If you insist that these are not "real life" categories, you might be more satisfied with the example of the category of fields, which has directed colimits but does not have coproducts or an initial object.

Consider any nontrivial group as a 1-object category. Then it has all filtered (co)limits (exercise: if all the morphisms in a filtered diagram are isomorphisms, then any object in the diagram is a (co)limit by taking an appropriate composition of the isomorphisms and their inverses). However, it does not have a (co)equalizer of any two distinct morphisms, or a (co)product of any number of copies of the unique object besides 1.

This is not really an answer, as I don’t know examples, but I think I might have a reason for why they are considering them separately.

Filtered colimits (I always get confused over directed / inverse) are particularly nice in concrete categories like $\mathsf{Set}, \mathsf{Ab}, \mathsf{Mod}_R, \mathsf{Top}$ and alike. There is an explicit formula for computing and dealing with them in $\mathsf{Set}$, which lifts to similar formulas in other concrete categories. From this formula one can deduce for example that filtered colimits commute with finite products (only for good categories!), which does not hold for arbitrary colimits! They may have even more special properties.

Long story short, often we are not interested in dealing with arbitrary shapes of colimits but only want to work with nice ones like coproducts, quotients, pushouts, gluing constructions or filtered colimits, of which we might know more than just „they are colimits“.

Part of the reason might be as well that most people don’t want to be bothered with abstract nonsense, but rather like to work with these things implicitly...