Cohen and Kaplansky have proven that a commutative ring $R$ has the property

C: Every $R$-module is a direct sum of cyclic $R$-modules.

if and only if $R$ is an Artinian principal ideal ring. Can we classify those commutative rings $R$ with the following property?

C': Every $R$-module is a direct summand of a direct sum of cyclic $R$-modules.

Actually I am interested in the following property, which is more involved:

C'': For every $R$-module there is a commutative $R$-algebra $R'$ satisfying $[\forall T \in \mathsf{Mod}(R):~R' \otimes_R T = 0 \Rightarrow T=0]$ (for example, a faithfully flat algebra) such that the $R'$-module $R' \otimes_R M$ is a direct summand of a direct sum of localizations of cyclic $R'$-modules.

If $R$ is arbitrary, what can we say about this class of $R$-modules?