We will use the following terminology.
A compactly generated \(\infty\)-category is an \(\infty\)-category with small colimits, for which there exists a small set of compact objects which generates \(\mathcal C\) under small colimits.
A projectively generated \(\infty\)-category is an \(\infty\)-category with small colimits, for which there exists a small set of compact-projective objects which generates \(\mathcal C\) under small colimits.