In the language of centralizers, a balanced module is one satisfying the conclusion of the double centralizer theorem, that is, the only endomorphisms of the group M commuting with all the R endomorphisms of M are the ones induced by right multiplication by ring elements.
[1] It turns out that being balanced is a left-right symmetric condition on rings, and so there is no need to prefix it with "left" or "right".
This study was continued in V. P. Camillo's dissertation, and later it became fully developed.
In addition to these references, K. Morita and H. Tachikawa have also contributed published and unpublished results.
A partial list of authors contributing to the theory of balanced modules and rings can be found in the references.