According to nlab, a category $C$ is called locally presentable if it is accessible and has all small colimits.
Moreover, one can show, that this conditions are equivalent to the condition of $C$ being a reflective subcategory of some presheaf-category $PSh(K)$ for a small category $K$ such that the inclusion $i\colon C\hookrightarrow PSh(K)$ is an accessible functor.
I once saw the definition of $C$ being locally presentable if $C$ is just a reflective subcategory of some presheaf-category $PSh(L)$?
Are this conditions equivalent? If yes, do I have to change the category $K$ in order to get the last 'definition' (i.e. is the second definition just redundant and the inclusion is automatically accessible)?