Terminology The terms “$\kappa$-compactly generated (∞,1)-category” and “locally ∞-presentable (∞,1)-category” have the same meaning. There are differences in usage, though.

If we “drop the $\kappa$”, then a locally presentable (∞,1)-category is a an (∞,1)-category which is locally $\kappa$-presentable for some$\kappa$, but a compactly generated (∞,1)-category is an (∞,1)-category which is locally finitely presentable, i.e. locally $\kappa$-presentable for $\kappa = \aleph_0$.

If we “leave the $\kappa$ in”, the terms “$\kappa$-compactly generated (∞,1)-category” and “locally ∞-presentable (∞,1)-category” have the same meaning. Some authors choose one term over the other. For example, in Higher Topos Theory, “$\kappa$-compactly generated (∞,1)-category” is preferred. Albeit, Lurie uses $\mathrm{Pr}_\kappa$ to denote the (∞,1)-category of $\kappa$-compactly generated (∞,1)-categories.