With x-ray absorption spectroscopy and polarized neutron reflectometry we studied how the magnetic proximity effect at the interface between the cuprate high-$T_C$ superconductor $mathrm{YBa_2Cu_3O_7}$ (YBCO) and the ferromagnet $mathrm{La_{2/3}Ca_{1/3}MnO_3}$ (LCMO) is related to the electronic and magnetic properties of the LCMO layers. In particular, we explored how the magnitude of the ferromagnetic Cu moment on the YBCO side depends on the strength of the antiferromagnetic (AF) exchange coupling with the Mn moment on the LCMO side. We found that the Cu moment remains sizeable if the AF coupling with the Mn moments is strongly reduced or even entirely suppressed. The ferromagnetic order of the Cu moments thus seems to be intrinsic to the interfacial CuO$_2$ planes and related to a weakly ferromagnetic intra-planar exchange interaction. The latter is discussed in terms of the partial occupation of the Cu $3d_{3z^2-r^2}$ orbitals, which occurs in the context of the so-called orbital reconstruction of the interfacial Cu ions.