If the first system is homogeneous, then the zero vector is in the solution set of the system. (See the proof of Theorem HSC.)
Since the two systems are equivalent, they have equal solutions sets (Definition ESYS). So the zero vector is in the solution set of the second system. By Exercise HSE.T10 the presence of the zero vector in the solution set implies that the system is homogeneous.
So if any one of two equivalent systems is homogeneous, then they both are homogeneous.