According to Wikipedia ("Free module", 03-Mar-2019, https://en.wikipedia.org/wiki/Free_module) "In mathematics, a free module is a module that has a basis - that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules." The same definition is used in [Lang] p. 135: "By a free module we shall mean a module which admits a basis, or the zero module."
In the following, a free module is defined as the direct sum of copies of a ring regarded as a left module over itself, see df-frlm. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm (see lmisfree), the two definitions are essentially equivalent. The free modules as defined by df-frlm are also taken as a motivation to introduce free modules by [Lang] p. 135.