The standard book on locally presentable categories defines them as : cocomplete categories with a small set of $\lambda$-small objets generating objects of the category under $\lambda$-filtered colimits.
nLab has the same definition but drops the $\lambda$-filtered condition.
Are those definitions equivalent? Or does nLab use a kind of Vopenka principle?
The definitions are equivalent.
Let $\mathcal{C}$ be a cocomplete locally small category. Let $\kappa$ be a regular cardinal and let $\mathcal{G}$ be a small full subcategory of $\mathcal{C}$ satisfying these conditions:
Then $\mathcal{C}$ is a locally $\kappa$-presentable category. The proof is not so hard: let $\overline{\mathcal{G}}$ be the closure of $\mathcal{G}$ under $\kappa$-small colimits in $\mathcal{C}$; then $\overline{\mathcal{G}}$ is the full subcategory of $\kappa$-presentable objects in $\mathcal{C}$, $\overline{\mathcal{G}}$ is essentially small, and every object in $\mathcal{C}$ is the colimit of a filtered diagram in $\overline{\mathcal{G}}$. Indeed, given any small category $\mathcal{J}$, let $\overline{\mathcal{J}}$ be the category obtained by freely adjoining finite colimits to $\mathcal{J}$; then $\overline{\mathcal{J}}$ is filtered, and for any diagram $X : \mathcal{J} \to \mathcal{G}$, there is a canonical extension $\overline{X} : \overline{\mathcal{J}} \to \overline{\mathcal{G}}$ such that ${\varinjlim}_\mathcal{J} X \cong {\varinjlim}_{\overline{\mathcal{J}}} \overline{X}$.
Borceux proves a stronger version of the above claim in §5.2 of [Handbook of categorical algebra, Vol. 2].