We study the evolution of the low-temperature field-induced magnetic defects observed under an applied magnetic field in a series of frustrated amorphous ferromagnets (Fe_1-xMn_x)₇₅P₁₆B₃Al₃ (“a-Fe_1-xMn_x”). Combining small-angle neutron scattering and Monte Carlo simulations, we show that the morphology of these defects resemble that of quasi-bidimensional spin vortices. They are observed in the so-called “reentrant” spin-glass (RSG) phase, up to the critical concentration x_C≈ 0.36 which separates the RSG and “true” spin glass (SG) within the low temperature part of the magnetic phase diagram of a-Fe_1−xMn_x. These textures systematically decrease in size with increasing magnetic field or decreasing the average exchange interaction, and they finally disappear in the SG sample (x= 0.41), being replaced by field-induced correlations over finite length scales. We argue that the study of these nanoscopic defects could be used to probe the critical line between the RSG and SG phases.